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: taras@physics.uq.edu.au (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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