Author’s Accepted Manuscript
Accurate absolute measurements of the Ramanbackscattering differential cross-section of waterand ice and its dependence on the temperature andexcitation wavelength
Taras Plakhotnik, Jens Reichardt
PII: S0022-4073(16)30875-5DOI: http://dx.doi.org/10.1016/j.jqsrt.2017.03.023Reference: JQSRT5633
To appear in: Journal of Quantitative Spectroscopy and Radiative Transfer
Received date: 19 December 2016Revised date: 10 March 2017Accepted date: 10 March 2017
Cite this article as: Taras Plakhotnik and Jens Reichardt, Accurate absolutemeasurements of the Raman backscattering differential cross-section of water andice and its dependence on the temperature and excitation wavelength, Journal ofQuantitative Spectroscopy and Radiative Transfer,http://dx.doi.org/10.1016/j.jqsrt.2017.03.023
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Accurate absolute measurements of the Ramanbackscattering differential cross-section of water and iceand its dependence on the temperature and excitation
wavelength
Taras Plakhotnika,, Jens Reichardtb
aSchool of Mathematics and Physics, The University of Queensland, Brisbane QLD 4072(Australia)
bRichard-Aßmann Observatorium, Deutscher Wetterdienst, Am Observatorium 12,Lindenberg 15848 (Germany)
Abstract
Measurements of Raman backscattering spectra between −15◦C and 22◦C in liq-
uid water (including its supercooled state) and in polycrystalline ice (−35◦C to
0◦C) at two excitation wavelengths (407 and 532 nm) are presented. It is found
that the spectrum-integrated backscattering cross section of the 3400 cm−1 band
is about 1.2 times larger for ice in comparison to liquid water. The excitation-
wavelength dependence of the cross section in ice is very close to that in water. A
discontinuous change of the spectrum is observed upon phase transition at 0◦C.
The results are applicable to preliminary calibration of lidar systems designed
for water content surveys in the atmosphere.
Keywords: Raman backscattering, cross section, water, ice, phase transition
1. Introduction
Raman scattering in water is important for many applications but, in par-
ticular, for lidar studies of the atmospheric water cycle as its understanding
is critical for weather and climate modeling and prognosis. Instrument devel-
opment aimed at first at accurate and continuous observation of the humidity5
∗Corresponding authorEmail address: [email protected] (Taras Plakhotnik)
Preprint submitted to Journal of Quantitative Spectroscopy & Radiative Transfer, March 11, 2017
profile for experimental ease, relative abundance of atmospheric water in its
vapor phase, and straight-forward calibration [1, 2, 3, 4]. Later, measurement
of the condensed water phases became the research focus because it holds the
promise to directly determine the water content of clouds [5, 6, 7, 8, 9, 10].
With the most advanced water Raman lidars such as [10, 11] it is now possible10
to measure the Raman backscattering coefficients of atmospheric water vapor,
ice, and liquid water. While the humidity measurement is routinely calibrated
using radiosondes, no such opportunity to compare exists for water content
profiles. Consequently, water content has to be determined ab initio from the
backscatter coefficients, which requires accurate knowledge of the molecular Ra-15
man backscattering differential cross section in frozen and liquid water at the
excitation wavelength of the lidar for small ice particles or droplets.
Because Raman scattering in water depends on intermolecular interactions
(in particular, on the hydrogen bonding), theoretical calculations of the Raman
spectra are very difficult. Significant deviations between theoretical modeling20
and experiments are observed despite the use of adjustable parameters and
an extensive computing power. For bulk liquid water, the agreement between
theory and experiments is rather semi quantitative [12, 13, 14]. The situation
is even more complex for small water droplets and ice crystals because their
morphology (shape) affects the scattering [15, 16, 17, 18] unless their sizes are25
an order of magnitude smaller than the wavelength of light or so large that the
surface effects can be either ignored or accounted for (reflection from a large
flat surface is an example where the calculations are simple). Generally, the
morphology especially of the ice crystals can be very diverse. In icy samples,
the scattering is also affected by the birefringent nature of crystalline materials30
and tensor nature of the scattering in the crystals [19]. At normal conditions, ice
forms uniaxial crystals where the ordinary and extraordinary refractive indices
at 546 nm are no = 1.3105 and ne = 1.3119 respectively [20]. The difference
is small but far from being negligible. Because of the difficulty with theoretical
estimations and modeling, the backscattering cross-section should be measured35
accurately in a laboratory environment.
2
Despite of a very long history [21], the published values of the cross-section
deviate up to a factor of 5 from each other even for bulk liquid water [22] but
generally the accuracy of the most reliable values for this case has been esti-
mated to be about 20%. The relatively large uncertainty of the best available40
estimate has, perhaps, prevented water from being included in the list of photo-
metric standards for Raman spectroscopy [23]. The situation for bulk liquid has
improved recently with publication of the scattering cross-section measured at
532-nm excitation with 1.5% absolute accuracy [24]. Impurities in atmosphere
and water, if present, do not affect the scattering cross-section directly but may45
obscure the Raman signal. This can be mitigated by using spectrally resolver
Raman spectroscopy instead of measurements of spectrally integrated intensity
(see [25] and references therein).
In ice, the situation is more complex. Due to the difference ne−no ≈ 0.0014,
the path length required to achieve the phase shift of 180 degrees between the50
two orthogonally polarized beams is on the order of 0.1 mm. Therefore it is
likely that light will be depolarized inside a polycrystalline ice sample even if it
is only a few milimeters thick. This, in turn, will affect Raman scattering espe-
cially if the scattering is measured in the orthogonal geometry (perpendicular
to the exciting beam of light), which is historically the most frequently used in55
the laboratory measurements. In such a geometry, the rotation of polarization
of the exciting laser beam by 90-degree aligns it parallel to the direction in
which the scattering is observed. The induced-dipole emission in the direction
of the dipole axis is zero and therefore the detected scattering in the direction
orthogonal to the exciting beam is significantly reduced. This effect is hard to60
take into account quantitatively and is a possible reason for a reported 1.6 times
larger cross-section in liquid water than in ice [26]. In the case of backscatter-
ing, the rotated polarization stays perpendicular to the scattering direction and
therefore the backscattered intensity will not be affected by the birefringence
in the first approximation. For this reason, the measurements of the scattering65
cross-sections will be more reliable in the backscattering geometry, also the main
mode of operation for lidar systems. By preparing ice samples of good optical
3
quality, we have also rectified the use of Raman spectroscopy for detecting the
water-ice phase transition. In our experiments, the transformation of the Ra-
man spectrum at the phase transition shows a discontinuity instead of about70
3◦C-wide transient as observed in [27].
This paper starts a series of experimental and theoretical publications aiming
at clarifying Raman scattering of different forms of water in atmosphere and
continues our previous work on accurate measurements of Raman scattering in
bulk water [24].75
2. Experimental
A simple confocal measurement scheme described earlier [24] has been used
also here. Such a design helps to deal with polarization dependent factors.
The measurements have been done at two excitation wavelengths, 532 nm (CW
laser, Verdi, Coherent) and at about 407 nm (CW diode laser, Thorlabs). The80
wavelengths are chosen so that they span a significant range of the excitation
wavelengths (a factor of 1.3 difference between the two) and the corresponding
Stokes regions (the water band is shifted by about 3400 cm−1 from the pump-
ing light) are covered by good luminescent standards of the quantum yield [28].
Most details of the experimental setup have been described in [24]. A new fea-85
ture essential for the measurement of the Raman spectra of ice and supercooled
liquid water is a home-designed microscope chamber where the samples can
be cooled from room temperature down to −35◦C with a carefully controlled
cooling rate.
2.1. Sample preparation90
The preparation of high-quality ice samples involves two main steps. First,
water (UltraPure, Invitrogen) is fast cooled down to approximately −18◦C to
form ice.These low-quality and milky samples are then heated up to about 1◦C
to start melting. Note that supercooled liquid water can be easily obtained
and maintained in our cold chamber for a long time in the temperature range95
4
between 0◦C and −15◦C. The moment only a few tiny crystals of the ice are left
in the sample chamber, the temperature is reduced again to about −0.2◦C to
start slow formation of the crystal phase from the ice nuclei. The dynamics of
this process at constant temperature of −0.15◦C is illustrated in Fig. 1. After
43 minutes, the first signs of the growing solid phase can be identified as a small100
rise of the left wing of the Raman spectrum. The feature, located at about 639
nm (marked by the arrow), is clearly visible at 48 min and growing thereafter.
Note that 6 more spectra have been taken between 50 min and 69 min (not
shown in this figure), which are identical to the ones at 50 min, and 69 min
and indicate that the crystal formation is complete. The spectrum taken at 43105
min can be decomposed into a sum of water (spectrum at t = 5 min) and ice
(spectrum at t = 69 min) contributions with weighting factors 0.89 and 0.11
respectively. The final result is a transparent, good optical quality ice block
of about 2.5 mm thickness. Note that the onset of melting has been detected
at −0.1◦C according to the chamber thermometer reading which indicates that110
the temperature of the sample has been correctly measured, with less than
−0.1◦C systematic error. This small offset has been subtracted from the sensor
readings at all temperatures. The accuracy of the sensor as specified by the
manufacturer is ±0.25◦C in the entire range of the temperatures used in these
experiments. The stability of the temperature during the measurement was very115
high. Generally the temperature fluctuated by approximately ±0.01◦C. Three
samples of ice were prepared. For each sample, spectra were measured with
two orthogonal orientations of polarization, SH and SV (vertical and horizontal
respectively). The average S = (SH+SV)/2 has then been analyzed as described
in Section 3.120
2.2. Spectrum calibration
Luminescence of Rhodamine 6G (R6G) in ethanol dissolved at a concentra-
tion of about 10−8M, excited at 532-nm wavelength and perylene in cyclohexane
(10−6M in cyclohexane, 407-nm excitation) have been used to obtain absolute
values of the scattering cross-sections (see [24] for details). The quantum yield125
5
40
80
120
0
2000
4000
6000
0
1
2
3
4
5
40
80
120
0
2000
4000
6000
-1
0
1
2
3
4
Ver
tical
coo
rdin
ate
40
80
120
0
2000
4000
6000
Spe
ctra
l int
ensi
ty o
f Ram
an s
catte
ring
(arb
. uni
ts)
-1
0
1
2
3
4
40
80
120
0
2000
4000
6000
0
1
2
3
4
5
Wavelength coordinate50 100 150
40
80
120
0
2000
4000
6000
Wavelength (nm)620 625 630 635 640 645 650 655 660 665 670
0
1
2
3
4
5
t=48 min
t=50 min
t=69 min
t=5 min
t= 43 min
×5
Figure 1: Formation of a polycrystalline ice sample of high optical quality. Images of raw
data from the CCD detector (left panels) and associated Raman backscattering spectra (right
panels) for different times after setting the temperature to −0.15◦C. The times are shown
in the legends. The 43-min panel highlights the difference between the 43-min spectrum and
5-min spectrum (wavy line at the bottom) and the difference between the 43-min spectrum
and a supposition of the 5-min spectrum and the 69-min spectrum (nearly straight line at
the bottom obtained using weighting factors of 0.89 and 0.11 respectively). Both differences
are multiplied by 5 for a better visibility. The 48-min panel also includes analogous difference
spectra with the weight factors of 0.24 and 0.76 for water and ice contributions.
6
values used in the calibration procedure have been taken as 0.950 ± 0.005 for
R6G [29] and 0.94±0.05 for perylene. Note that, for example, a decrease of the
yield from 0.94 to 0.90 makes the estimate for the cross-section proportionally
5% smaller. The relatively large uncertainty of the quantum yield of perylene is
the main source to the uncertainty of the scattering cross-section and is briefly130
discussed in the following.
At ambient partial pressure of oxygen, the luminescence quantum yield of
R6G is not affected significantly by the oxygen dissolved in ethanol [29] while
luminescence of perylene may be appreciably quenched. The reported quenching
by oxygen constant of perylene in dodecane is 112 M−1[30]. Given the solubility135
of oxygen in cyclohexane near room temperature is 1.16 × 10−3 mole fraction
atm−1 [31], the oxygen quenching of perylene in cyclohexane should decrease
the yield by a factor of about 1.25 at the ambient partial pressure of 0.21
atm−1 (assuming that the quenching constants in dodecane and cyclohexane are
similar). But a much smaller 5%-increase upon deaeration has been observed140
experimentally [32]. Nevertheless oxygen related quenching is still a factor of
concern because when perylene is dissolved in ethanol or benzene, the reported
change of the yield upon deaeration has been about 1.3 [33], in agreement with
the above estimate. This sheds some doubts on the 1.05 factor for cyclohexane
reported in [32]. To reduce oxygen concentration, our sample of cyclohexane has145
been placed in a helium atmosphere. This has resulted in a moderate increase
(a factor of 1.15) of the luminescence intensity which has been explained by
decreasing of the oxygen related quenching. Note also that the uncertainties for
the luminescence quantum yield in perylene have not been always thoroughly
investigated (for example, reported 0.99±0.03 [34] is logically inconsistent as the150
yield can not be larger than 1) and/or clearly stated in the available literature.
To get a better idea about the errors, we have compared the quantum yield
reported for a number of solvents – cyclohexane (0.94 [35] and 0.93± 0.09 (0.98
after deaeration) [32]), ethanol ( 0.93 ± 0.03[34] and 0.87 [33]), and benzene
(0.89 [33] and 0.99 [34]).155
Optical densities of the dye molecules in the corresponding solvents [24]
7
Wavelength (nm)400 450 500 550 600 650
Lum
ines
cenc
e in
t. (a
b.u.
)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Figure 2: Comparison of luminescence spectra measured on commercial spectrometer (blue
line) and in these experiments (red line). The difference is hard to see as it is smaller than
3%.
have been measured with about 1% accuracy using a commercial UV-Vis ab-
sorption spectrophotometer, a Varian Cary 5000. The wavelength dependent
transmissions of the optics, spectrometer, and sensitivity of the CCD have been
determined using a spectral lamp (Optronic Laboratories model: OL 200M) as160
a standard for spectral irradiance with less than 0.5% errors for relative values
of the irradiance over a wide wavelength range covered in our experiments. The
objective (Nikon, WD = 7 mm, NA = 0.55) collects a cone of the scattered
light with an opening angle 2Θ such that Θ = arcsin(NA). The calibration has
been validated by comparison of the luminescence spectra of our two references165
to their luminescence spectra measured with a commercial spectrometer Horiba
Jobin-Yvon Fluoromax 4. The spectra are shown in Fig. 2. The standard
deviation between the two curves is about 3%.
3. Analysis
Just as in [24], we consider a geometry where the exciting laser light, ini-170
tially polarized along the laboratory x-axis travels in the direction of the y-axis
through a bulk sample. For simplicity, we assume that this polarization rotates
due to the sample birefringence around the y-axis when the beam propagates
through the polycrystalline sample and neglect a possibility that the birefrin-
8
gence and refraction will result in a significant polarization in the y-direction.175
Thus, at any location inside the sample, the polarization of the laser beam is
in the xz-plane. Given that the sample is polycrystalline it can be seen as ap-
proximately homogeneous, and its Raman scattering can be presented (similar
to the case of liquid water) by three dipoles. One dipole is parallel to the local
polarization of the laser beam (we will call it x′-direction and note that it is not180
the same as the initial polarization of the laser beam x), another dipole points
along y, the direction of the beam propagation, and the third is orthogonal to
the first two and is parallel to z′-axis. The differential scattering cross-section
associated with these dipoles reads
dσ
dΩdλ= F [
px′(λ) sin2 θx′ + py(λ) sin2 θy + pz′(λ) sin2 θz′
](1)
The factor F absorbs all the units dependent constants while the angles θx′,y,z′185
are the angles between the corresponding axis (indicated by the subscripts)
and the direction of scattering. Similar to the liquid water case, we assume
that py(λ) = ρ(λ)px′(λ) and pz(λ) = ρ(λ)px′(λ), where ρ(λ) is the wavelength
dependent depolarization factor. The backscattering cross-section (θx′ = θz′ =
90◦ and θy = 0◦) is then190
dσb
dΩdλ= Fpx′(λ) [1 + ρ(λ)] (2)
The spectral photon density S(λ) intercepted by the microscope objective (this
is the signal recorded by the photon counting CCD) reads
S(λ) = D(λ)Fpx′(λ)[Π⊥(1 + ρ(λ)) + Π‖ρ(λ)
](3)
where D(λ) is the photon detection efficiency. The geometrical factors Π⊥ and
Π‖ are defined in [36] as follows
Π⊥ =1
32[16− 15 cos(θ)− cos(3θ)] (4)
Π|| =1
16[cos(3θ)− 9 cos(θ) + 8] (5)
where sin(θ) = sin(Θ)/n, n being the refractive index of water (liquid or solid).195
Generally, the radiation of the three emitting dipoles has different polarization
9
but the detected signal S(λ) here is defined as an average of spectra measured
with vertically and horizontally polarized laser light. Due to the axial symmetry
of the setup the averaged spectra are effectively depolarized and this makesD(λ)
insensitive to polarization. Finally, S(λ) can be related to the backscattering200
cross-section as followsdσb(λ)
dΩdλ=
S(λ)G(λ)
D(λ)(6)
where the geometrical factor G(λ),
G(λ) =1 + ρ(λ)
Π⊥ (1 + ρ(λ)) + Π‖ρ(λ), (7)
takes into account the effect of the numerical aperture and the depolarization
factor on the detected signal for a given value of the backscattering cross-section.
The factor D(λ) can be obtained experimentally by using luminescence stan-205
dards with known quantum efficiency for calibration. In actual lidar experi-
ments, the value of the NA is much smaller than 1 and therefore Π⊥ � Π‖
and hence G = 1/Π⊥. For the case of NA = 0.55, Π⊥ = 0.0667, Π‖ = 0.0063,
the value of G varies between 14.86 and 14.35 (that is by about 3%) when the
value of ρ changes from 0.1 to 0.9. Although ρ(λ) is difficult to determine ex-210
perimentally for the ice sample due to its birefringence, this uncertainty does
not affect a lot the value of the scattering cross-section as the estimate above
shows. For the purpose of calculations of G(λ), we have assumed ρ(λ) to be the
same as in the case of liquid water (the assumption is somewhat justified by the
polycrystalline nature of the samples).215
4. Results and Discussion
The experimental results are shown in Figs. 3–5. All spectra are intensity-
calibrated using the luminescence standards described in Section 2. Figure 3
shows the temperature dependence of Raman scattering in both water phases at
an excitation wavelength of 532 nm, with the observed trends being highlighted220
in Fig. 4. Figure 5 focuses on the dependence of the Raman backscattering
spectrum on excitation wavelength.
10
Bac
ksca
t. sp
ectru
m (c
m2 n
m-1
sr-1
) ×10-31
0
1
2
3
4
5 -34.6oC-22.6oC -0.2oC-14.6oC (L) -4.6oC (L) 20.4oC (L)
Wavelength (nm)625 630 635 640 645 650 655 660 665
ρ(λ
) for
liqu
id
0
0.1
0.2
0.3
-14.6°C 0.4°C 20.4°C
Figure 3: Top panel: Examples of differential backscattering cross-sections in liquid water
(L) and ice measured at 532-nm excitation and different temperatures (as indicated in the
legend). Bottom panel: Depolarization ratio for liquid water at several temperatures (see the
legend). The curve for −4.6◦C lies in between the 0.4◦C and −14.6◦C curves.
The first observation that can be made is that in our experiments the sig-
nature in the Raman spectra of the water-ice phase transition is much sharper
than previously reported [27], as evidenced in Fig. 4A. The relative intensity at225
two reference wavelengths in the spectrum (analogous to the one used in [27]),
the symmetric stretching mode centered near 3160 cm−1 and the asymmetric
stretching at 3410 cm−1 displays a discontinuity as expected for first-order tran-
sitions. Unlike the results of [27], the transition in our sample is much more
definitive on the temperature scale (does not show any transient states). The230
spectral difference between liquid and solid phases abruptly appears as soon
as the solid phase is formed. Probably the samples studied in [27] contained
a mixture of water and ice and were not at thermodynamical equilibrium over
a range of set temperatures. This explanation suggests that the temperature
control and homogeneity of the sample are much improved here in comparison235
to [27].
Notably, the relative strength of the Raman scattering at these two points
in the spectrum does not change significantly in the solid phase. This suggests
11
that the ratio can be considered as a classical disorder parameter. The disorder
parameter, a counterpart of order parameter is zero in the ordered phase and240
nonzero in the disordered phase but has a discontinuity upon the phase tran-
sition (see about the order parameter in [37]). The ratio shows a linear trend
of 0.015 K−1 for the liquid phase and thus indicates that some partial order-
ing is identified above the phase transition. This ordering can be associated
with increasing probability of formation of long lasting hydrogen bonds upon245
cooling [38]. The variation of the spectrum of ice with the temperature is most
apparent for a different pair of spectral points as shown in Fig. 4B. These two
reference points measure a shift of the symmetrical mode towards a lower fre-
quency when the temperature decreases. The trend is also approximately linear
but has a smaller gradient of about 8.5× 10−3 K−1. Both dependencies can be250
used to measure temperatures of H2O in the corresponding phase state.
Finally, we show in Fig. 4C the temperature dependence of the integrated
backscattering cross-section. Again, a clear discontinuity is observed near the
phase transition. The reason for the sudden change in the cross-section upon
the phase transition is explained by local field effects and by redistribution of255
electron density due to intermolecular coupling [39]. The local field factor can
be estimated [40] using the Lorentz -Lorentz model where the scattering cross-
section scales as followsdσ
dΩ∝ 1
(n2 + 2)4. (8)
This factor is about 9% larger for ice (n ≈ 1.31) in comparison to water (n ≈1.34). The cross-section actually increases several times upon the transition260
from gas to the liquid phase (the integrated cross-section of vapor is reported
to be (1.10 ± 0.12) × 10−30cm2sr−1 at 515-nm excitation [41]). The observed
independence of the integrated cross-section on the temperature (for the same
phase) is in an apparent disagreement with published results [40, 42] reporting
an increase of about 9% from room to−15◦C for liquid water. This disagreement265
is due to the fact that Refs. [40, 42] compare so-called isotropic contributions
to the spectra expressed as SVV − 43SVH, where the signals are measured in
12
-40 -30 -20 -10 0 10 20 30
dσb/d
Ω (c
m2 /s
r)
×10-30
5.65.8
66.26.46.66.8
Temperature (°C)
-35 -30 -25 -20 -15 -10 -5 0
σb(6
38.1
)/σb(6
41.1
)
0.9
1
1.1
1.2
1.3
-40 -30 -20 -10 0 10 20 30σb(6
50.3
)/σb(6
39.4
)
0
0.5
1
1.5
2(A)
(B)
(C)
Figure 4: Dependence of the Raman spectra of H2O (excitation at 532 nm) on the tempera-
ture of the sample. Panel A (circles) shows the ratio of the scattering intensities measured at
650.3 nm (≈ 3410-cm−1 Stokes shift) and 639.4 nm (≈ 3160 cm−1). The upper curve presents
data for the liquid phase (including the supercooled water below 0◦C) and the bottom curve
for ice. The middle curve is the data from [27]. Panel B shows the ratio of the scattering
intensities measured at 638.1 nm (≈ 3125 cm−1) and 641.1 nm (≈ 3190 cm−1) in the solid
phase. Panel C displays the dependence of the backscattering cross-section on the tempera-
ture and a discontinuous jump upon the phase transition (the bottom curve corresponds to
liquid).
13
2800 3000 3200 3400 3600 3800
Diff
eren
tial b
acks
catte
ring
(cm
2 nm-1
sr-1
)
×10-31
00.5
11.5
22.5
33.5
44.5
5
spec
trum
diffe
renc
e
532 nm, 22°C407 nm, 22°C532 nm, 2°C407 nm, 2°C532 nm, -2.6°C407 nm, -1.5°C
Stokes shift (cm-1)2800 3000 3200 3400 3600 3800
×10-32
-1
0
1
2
3
Figure 5: Differential Raman backscattering. Top panel: Differential cross-section spectra
in liquid water and ice at different temperatures and excitation wavelengths. The horizontal
scale is in wavenumbers Stokes shift to facilitate the comparison (as indicated in the legend).
The spectra measured at 407 nm are divided by 7.5, the factor equalizing the integrated
cross-sections obtained at 22◦C with 407 and 532 nm excitation. Bottom panel: Absolute
differences (note the change of the vertical scale) between the corresponding spectra (after
the 7.5 scaling).
14
the 90◦ geometry. The wavelength integrated isotropic contribution is then
∝ ∫px(λ)[1− 4
3ρ(λ)]dλ. The weighting factor of [1− 43ρ(λ)] emphasizes the rise
of the spectrum with decreasing temperature to the left from 647.5 nm where a270
relatively smaller value of ρ(λ) (see Fig. 3) makes the factor larger.
All the results for the Raman backscattering cross-section in liquid water
and ice are summarized in Table 1. The integration is done over the same
frequency range regardless of the excitation wavelength (the range of Stokes
shifts is between 2670 cm−1 and 4510 cm−1). For both phases, extrapolated275
values are determined by using the empirical expression published in [22] for
liquid water. Cross-sections measured at 532 nm are extrapolated to 407-nm
excitation, and the measurement results obtained at 407nm are extrapolated to
355nm, the common transmitter wavelength in Raman lidars. The expression
used for extrapolation reads280
dσ
dΩ= A
ν4s(ν2i − ν2p)
2(9)
where νi = 88000 cm−1 is the frequency of an effective intermediate resonance,
νs is the frequency of the scattered light, νp is the frequency of the pump light
(laser), and the constant A can be defined so that the value of the cross-section
agrees with the data measured at the reference wavelength. Comparison of the
measured and extrapolated values at 407 nm shows a disagreement of less than285
10% for both ice and water. Thus it can be inferred that the relation is not only
applicable to liquid but also to frozen water. The estimated accuracy of Eq.(9)
over a large (215–550 nm) range of wavelengths is about 20% [22]. We expect
that the accuracy of extrapolation from 407 nm to 355 nm (where the change
of the cross-section is a factor of 2 smaller than when 532 nm and 407 nm are290
compared) will be better than 10% but a conservative estimate of 10% error is
included in the table as the error of the extrapolation. The error bars of the
measured values cover the fluctuation of the cross-section values measured at
different temperatures (see Fig.4).
15
Table 1: Differential backscattering cross-section of H2O in different phase state and for
different excitation wavelengths (measured and estimated)
Integrated differential Raman backscattering
Excitation wavelength cross-section in units of 10−30cm2/(sr molecule)
nm Liquid Ice
532 (measured) 5.8± 0.1 6.6± 0.2
407 (extrapol.) 22 25
407 (measured) 23.3± 1.4 27.4± 2
355 (extrapol.) 46± 5 54± 5
5. Conclusion295
We have measured the differential cross-section spectra of Raman backscat-
tering in liquid water between −15◦C and 22◦C (including supercooled water),
and in ice between −35◦C and 0◦C at two excitation wavelengths, 532 nm and
407 nm, which enabled us to study their temperature and wavelength depen-
dence in detail. The temperature dependence shows a discontinuity, a clear300
signature of a phase transition near 0◦C. The temperature dependence of
the spectra is explained by rearrangement of hydrogen bonding between wa-
ter molecules and can be used for phase transition identification and for tem-
perature measurements of water and ice. The dependence of the differential
backscattering in ice on the excitation wavelength is similar to the dependence305
found for liquid water in previous studies. This finding allows for an easy and
reliable extrapolation of the scattering cross-section also in ice to the wave-
lengths not covered in these experiments. At 355 nm, a common transmitter
wavelength of lidars targeting Raman scattering in atmospheric water, the two
condensed phases are determined to have the following differential cross-sections310
of Raman backscattering: (4.6 ± 0.5) × 10−29 cm2sr−1molecule−1 (liquid) and
(5.4± 0.5)× 10−29 cm2sr−1molecule−1 (ice). The results are applicable to pre-
liminary calibration of these lidars. One has to note, however, that the cross-
16
sections determined in this study are for bulk water and ice and may not be
fully representative for tiny water droplets and ice microcrystals. Therefore315
more research is needed to quantify the significance of the particle shape and
size effects on Raman backscattering and thus to fully establish direct measure-
ments of cloud liquid and ice water content.
6. Acknowledgements
This research has been supported in part by ARC grant DP0771676.320
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