The Refractive Index of Amorphous and Crystalline Water Ice in the UV–vis

The refractive index n can be measured during the deposition of water vapor on a reflective or transparent surface. For volatile species, this is usually done on the tip of a liquid nitrogen or closed cycle helium cryostat placed inside a vacuum chamber (e.g., Goodman 1978; Brown et al. 1996; Dohnálek et al. 2003).

As the thickness increases during the ice growth, periodical fluctuations in the intensity of the transmitted or reflected light appear. These fringes occur due to constructive and destructive interference arising from a difference in the path length of light reflecting internally in the ice and light transmitting directly. The refractive index of the sample is related to the period of the fringes and the thickness d by

$$\begin{eqnarray}&&d=\displaystyle \frac{m\lambda }{2n(\lambda )\cos \theta },\end{eqnarray} \tag{ 1 }$$

where m is the total number of fringes observed at wavelength λ (in nm), n(λ) is the wavelength-dependent refractive index, and θ is the angle of incidence with respect to the surface. Interference is commonly measured in reflection (e.g., Baratta & Palumbo 1998; Fulvio et al. 2009) but can also be observed in transmission. Interference measurements are typically performed using a monochromatic light source, such as a helium–neon laser. The strength of our new approach is that we simultaneously record the interference for all wavelengths covered by a broadband light source. The details are given in Section 2.2.

Warren (1984) and Warren & Brandt (2008) reported n(λ) of crystalline water ice between 200 and 400 nm at a resolution of 50 nm, and between 400 and 800 nm at a resolution of 10 nm. In these cases, discrete wavelengths are measured in separate experiments, placing practical limits on the full wavelength coverage and resolution and ultimately limiting the precision of n to how well the experiments are reproduced. In the case of ASW, n is predominantly measured using helium–neon lasers, and therefore reported at a wavelength of 632.8 nm (Brown et al. 1996; Westley et al. 1998; Dohnálek et al. 2003; Romanescu et al. 2010). Bouilloud et al. (2015) provided an overview of known values of the refractive index for a number of other astrophysically relevant ices, including H₂O, CO₂, CO, CH₄, and NH₃.

The refractive index depends on the density (porosity) of the solid. Vacuum deposition of crystalline ice results in a constant density, but for ASW, the porosity increases at lower deposition temperatures (see, e.g., Brown et al. 1996; Westley et al. 1998; Dohnálek et al. 2003). The porosity and the refractive index are related via the Lorentz–Lorenz equation:

$$\begin{eqnarray}&&R(\lambda )=\displaystyle \frac{1}{\rho }\displaystyle \frac{n{\left(\lambda \right)}^{2}-1}{n{\left(\lambda \right)}^{2}+2}.\end{eqnarray} \tag{ 2 }$$

Here, R(λ) is the specific refraction (in units of cm³ g⁻¹) and ρ is the density of the ice. We adopt R(632.8) = 0.2072 cm³ g⁻¹ (Brown et al. 1996) at a density of 0.93 g cm⁻³ and assume that R is constant at a given wavelength, regardless of the morphology of the ice.³ We will verify the latter assumption. In order to determine the refractive index of porous ASW, one needs to determine or assume the level of porosity. For this study, we rely on the ASW densities reported as function of the deposition temperature by Dohnálek et al. (2003). Using Equation (2) together with the adopted R-value yields the refractive index.

The experiments were done using a recently constructed setup, described in detail in Kofman et al. (2018). We will briefly describe the experimental features that are relevant to this study. The setup is a high vacuum system with a base pressure in the order of 5 × 10⁻⁸ mbar. Residual gas in the chamber is dominated by water. At the center of the chamber, a BaF₂ sample window is mounted onto the cold-finger of a closed cycle helium cryostat. The temperature of the sample is measured using a chromel-Au/Fe thermocouple and controlled with resistive heating. Temperatures between 10 and 320 K are accessible with an absolute accuracy better than 2 K. For these experiments, water ice is grown using background deposition, the same method as used by Dohnálek et al. (2003). We use a precision dosing valve, which results in a constant deposition rate with less than 3% variation in the pressure over the entire deposition time, and also allows reproducible water partial pressures inside the chamber between different experiments, ensuring that the conditions of separate experiments are comparable.

During water deposition, the chamber pressure is 2.6 × 10⁻⁶ mbar, corresponding to a deposition rate of about 0.3 nm s⁻¹ or roughly 1 mono-layer per second. Experiments were performed with sample temperatures of 10, 30, 50, 70, 90, 110, 130, and 150 K. Only in the latter case does this result in the formation of crystalline ice; all other experiments concern ASW.

UV–vis spectra are continuously recorded during the deposition. For the work performed here, the spatially filtered light of a broadband Xe-arc lamp, emitting in the range of 180–1100 nm, is guided through the sample that is positioned perpendicular to the light beam. Light is dispersed by an Andor 303i Shamrock spectrometer equipped with a 10 μm slit and collected on a thermo-electrically cooled CCD (Andor iDus DV420 OE). We record data from 210 to 757 nm at a spectral resolution of 0.56 nm, and at a time resolution of 4.3 s. The broadband spectroscopy applied allows us to measure interference fringes over the full wavelength range during the deposition. From the interference fringes, n can be determined as a function of λ in one experiment, something not possible when only monochromatic light sources are used.

The use of background deposition causes ices to grow on both the front and back side of the window. In the ideal case, the ratio of the deposition rates on each side of the sample would be unity (i.e., $\tfrac{{\phi }_{1}}{{\phi }_{2}}=1$, with ϕ_i being the deposition rate at the respective side). In practice, this ratio varies between 0.91 and 0.96 in the experiments performed in this study. Measuring the interference pattern monitors the cumulative effect of similar but not fully identical growth processes on both sides of the window. As will be shown below, the effect of asymmetric deposition can be corrected by taking into account the empirically determined $\tfrac{{\phi }_{1}}{{\phi }_{2}}$-value. As the light propagates through the growing ice on both sides of the deposition window, the absolute light intensity I at wavelength λ fluctuates as a function of the time t. When the deposition rates at both sides are equal, the signals can be described by a single periodic function:

$$\begin{eqnarray}&&I=A\cos \left(\displaystyle \frac{2\pi }{P(\lambda )}t\right).\end{eqnarray} \tag{ 3 }$$

P(λ) is the wavelength-dependent period, and A is the magnitude of the oscillation. Several studies focus on the absolute values of A, and we refer the reader to these reports for a full description of the amplitude of the signal (Goodman 1978; Dohnálek et al. 2003; Romanescu et al. 2010). Note that as in our experiments the light progresses at a normal incidence, the $\cos \theta$ term in Equation (1) equals unity and is left out.

In the case of asymmetric deposition, two cosine terms, one for each side of the window with a corresponding period P(λ) and amplitude A, are required to reproduce the observed interference pattern.

$$\begin{eqnarray}&&I={A}_{1}\cos \left(\displaystyle \frac{2\pi }{{P}_{1}(\lambda )}t\right)+{A}_{2}\cos \left(\displaystyle \frac{2\pi }{{P}_{2}(\lambda )}t\right).\end{eqnarray} \tag{ 4 }$$

Note that we cannot discriminate which part of this equation describes which side of the window. In order to determine the period from the experimental signals, we take A₁ = A₂ = $\tfrac{1}{2}A;$ i.e., both sides of the window contribute equally to the amplitude of the signal. This allows us to fit a single value of A, which significantly reduces the complexity of the fitting routine. Similarly, the period is fit with a single value for both signals in the first iteration. Subsequently, we introduce the $\tfrac{{\phi }_{1}}{{\phi }_{2}}$ factor in the second cosine, substituting P₂(λ) with $\tfrac{{\phi }_{1}}{{\phi }_{2}}$ P₁(λ). This comes from the fact that the period P(λ) is directly related to the deposition rate ϕ (see Equation (6)). The $\tfrac{{\phi }_{1}}{{\phi }_{2}}$ factors are constrained manually for each experiment and are included in the final fits that start from the best-fit solution of the first iteration. Table 1 shows the values of $\tfrac{{\phi }_{1}}{{\phi }_{2}}$ used for the fits of all the experimental temperatures. As the occurrence of the interference minima (see below) is fully determined by $\tfrac{{\phi }_{1}}{{\phi }_{2}}$, we can constrain $\tfrac{{\phi }_{1}}{{\phi }_{2}}$ quite accurately. Based on this, we estimate the error on the value for $\tfrac{{\phi }_{1}}{{\phi }_{2}}$ in the order of 1%.

Table 1.  Density of Background-deposited ASW Reported by Dohnálek et al. (2003), the Total Thickness d, $\tfrac{{\phi }_{1}}{{\phi }_{2}}$, σ_fit, Which is the Standard Error on the Refractive Index from the Fit of Equation (7) and Best-fit Values for B₁ and B₂ from Section 3.3

Temperature (K)	ρ	d (nm)	$\tfrac{{\phi }_{1}}{{\phi }_{2}}$	n(632.8) ± σfit	B1	B2
10	0.585 ± 0.010	1703.6 ± 6.0	0.94 ± 0.01	1.190 ± 0.004	0.309	0.098
30	0.636 ± 0.010	1659.3 ± 9.7	0.94 ± 0.01	1.208 ± 0.007	0.374	0.099
50	0.688 ± 0.010	1559.8 ± 4.3	0.94 ± 0.01	1.226 ± 0.003	0.331	0.164
70	0.740 ± 0.010	1481.7 ± 2.2	0.94 ± 0.01	1.244 ± 0.002	0.377	0.164
90	0.791 ± 0.010	1436.1 ± 5.9	0.95 ± 0.01	1.262 ± 0.003	0.280	0.291
110	0.843 ± 0.010	1343.2 ± 3.2	0.96 ± 0.01	1.280 ± 0.003	0.400	0.220
130	0.895 ± 0.010	1317.6 ± 2.2	0.96 ± 0.01	1.299 ± 0.002	0.417	0.258
150	0.925 ± 0.010	1177.6 ± 5.7	0.92 ± 0.01	1.310 ± 0.006	0.496	0.190

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The experiments at 90 and 10 K represent two different cases of asymmetric deposition, with $\tfrac{{\phi }_{1}}{{\phi }_{2}}$ = 0.96 and 0.94, respectively. Figure 1 shows the interference signal measured during the deposition at a few selected wavelengths throughout the recorded range. The seven wavelengths are picked at intervals of 80 nm across the entire measured range. The left panel shows the deposition at 90 K, and the right panel at 10 K. The resulting fits using Equation (4) are also shown. The effect of the asymmetric deposition rates is seen in the decrease in the amplitude of the interference signal after roughly 10 periods in the left panel and after 6 periods in the right panel. Note that this lapse in intensity is persistently seen after the same number of periods at the different wavelengths. Currently, we do not have a full explanation for the difference in the deposition rates at the two sides of the sample, but we intend to make this the subject of an upcoming study. Finally, note that the signal becomes more noisy at wavelengths above 400 nm, an effect of small intensity fluctuations in the spectral energy distribution of the xenon light source.

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In order to calculate the refractive index from the period, we substitute d = ϕmP(λ) in Equation (1). This comes from the fact that the thickness of the ice is equal to the deposition rate ϕ, multiplied by the total deposition time, mP(λ).

This yields:

$$\begin{eqnarray*}&&\phi {mP}(\lambda )=\displaystyle \frac{m\lambda }{2n(\lambda )}\end{eqnarray*}$$

and after reordering:

$$\begin{eqnarray}&&n(\lambda )=\displaystyle \frac{\lambda }{2P(\lambda )\phi }.\end{eqnarray} \tag{ 5 }$$

We use this equation to express n_e(λ), with the subscript e indicating that these values are obtained from the experiments, as a ratio to n(632.8):

$$\begin{eqnarray}&&\displaystyle \frac{{n}_{e}(\lambda )}{n(632.8)}=\displaystyle \frac{P({\lambda }_{632.8})}{P(\lambda )}\displaystyle \frac{\lambda }{632.8}.\end{eqnarray} \tag{ 6 }$$

We determine n(632.8) from the Lorentz–Lorenz equation (Equation (2)), and we use the specific refraction R = 0.2072 cm³ g⁻¹ and the density ρ of background-deposited ice from Dohnálek et al. (2003), using their Figure 8 and fitting a linear function for the densities derived between 22 and 140 K. We follow Dohnálek et al. (2003) in their assumption that the level of porosity depends linearly on the deposition temperature (for the range studied). The densities and the resulting refractive indices are shown in Table 1.

Using Equation (6) and the periods from the fits of the UV–vis signal recorded during the ice deposition, the values of n_e(λ) are obtained. The left panel of Figure 2 shows n_e(λ) obtained from the signal recorded during the deposition of ASW at 90 K. The figure shows that n_e(λ) increases at shorter wavelengths. Between 400 and 760 nm, the refractive index exhibits oscillations of the order of 1% of the n-value, likely due to the lower signal-to-noise ratio at these wavelengths. We can reproduce the fringes and the overall interference pattern seen in the experimental signals when we substitute the period in Equation (4) and calculate the amplitude I on a wavelength-time grid, simulating the UV–vis interference pattern. This gives a qualitative impression of the performance of the fitting routines. In the middle panel of Figure 2, we show the simulated interference pattern, with the right panel showing the recorded UV–vis interference pattern. We see that the fitting routine reproduces the experimental signals well. Figure 3 shows the same panels for the experiment at 10 K. At 10 K, the destructive interference clearly results in a low-amplitude band after six periods, starting at roughly 2000 s at 200 nm, and falling outside of the experimental time range at wavelengths longer than 600 nm. This same pattern can be seen in Figure 1. The experimental signal also shows vertical bands, indicating fluctuations in the total lamp flux, which already were noticed at longer wavelengths in Figure 1.

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In the previous section, the refractive index derived from the experimental interference patterns showed relatively small and non-physical oscillations above 400 nm. By using a continuous function for n(λ), we can eliminate these artifacts. The wavelength dependency of the refractive index can be approximated by using the Sellmeier dispersion equation (Sellmeier 1871):

$$\begin{eqnarray}&&{n}_{S}{\left(\lambda \right)}^{2}=1+\displaystyle \frac{{B}_{1}{\lambda }^{2}}{{\lambda }^{2}-{C}_{1}}+\displaystyle \frac{{B}_{2}{\lambda }^{2}}{{\lambda }^{2}-{C}_{2}},\end{eqnarray} \tag{ 7 }$$

where each term in the summation represents an absorption resonance of strength B at wavelength $\sqrt{C}$, and n_S(λ) is labeled with the subscript S to indicate these values are derived using the Sellmeier equation. We use two sets of parameters to describe n_S(λ), corresponding to absorption maxima at 71 and 134 nm ($\sqrt{{C}_{1}}$ and $\sqrt{{C}_{2}}$), the vacuum UV absorption maxima of ASW (Cruz-Diaz et al. 2014). This relates back to the fact that the real and imaginary refractive indices are connected. We assume that the electronic absorption maxima do not change significantly when changing from amorphous to crystalline water ice, an assumption we later verify. B₁ and B₂ are fit using a least-squares approach with a linear cost function. The best-fit values for B₁ and B₂ are reported in Table 1. Note that although we adopted the refractive index at 632.8 nm to calculate n_e(λ), we do not force the Sellmeier fit through this value.

A conservative estimate of the relative error of the resulting function for n_S(λ) can be obtained by calculating the difference between the experimental fit of the period and the Sellmeier fit. The difference between n_S(λ) and the experimental n_e(λ) provides a measure for the uncertainty at this wavelength. The error, σ_λ, is calculated by taking the root mean square error of the two values of n at wavelength λ. The average is shown as σ_fit in Table 1.

In Figure 4, we show the refractive index from the period (from Section 3.1), together with the fit of Equation (7) for all the temperatures, including the 10 and 90 K measurements already shown in Figures 2 and 3. The experimental UV–vis patterns are shown as well. The graphs are shown in pairs of two, where the left panel shows n_e(λ) and n_S(λ), and the right panel shows the experimental signal. Clearly, the fit based on Equation (7) removes the oscillations still visible when using Equation (4).

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Although the Sellmeier fits in Figure 4 are generally well constrained, we see disagreements between the experimentally determined values for n_e(λ) and n_S(λ) in some wavelength domains, particularly in the cases of the 30 and 150 K experiments above 600 nm. Here, the interference patterns are relatively noisy, and as a result, the standard error σ_fit on the fits is larger for these two experiments. The uncertainty in the resulting n_S(λ) values can be further decreased by using the Lorentz–Lorenz equation and producing R_T(λ) for each temperature, where the subscript T indicates the deposition temperature, and using the average of the eight measurements to provide one set of values for R_a(λ). This in turn allows us to derive a general function for n(λ, ρ) as a function of wavelength and porosity, which has the advantage that we can calculate the refractive index for any ice porosity. Note that for this, we assume R_632.8 = 0.2072 cm³ g⁻¹ and R_a to be independent of temperature, and that the electronic absorption features of ASW and crystalline ice are not significantly different. The respective curves of R_T(λ) in Figure 5 for the amorphous and crystalline experiment indicate that this assumption is valid within the uncertainty of the experiments: the largest outlier in this figure is the R₉₀(λ) curve, and the R₁₅₀(λ) curve agrees closely with those from the other experiments. This indicates that the porosity is the governing factor in the refractive index and that the ordering of the water molecules in the solid phase (i.e., ASW or crystalline ice) is less important. Note that the use of the specific refraction to determine the refractive index of ASW is widely used in the literature (e.g., Brown et al. 1996; Dohnálek et al. 2003, and implicitly in the subsequent work adopting the refractive index from these studies). Here, we provide the proof that this approach is indeed valid.

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A continuous function for n(λ, ρ), i.e., explicitly using the density ρ and based on the derived R_a(λ) curve, is obtained by rewriting Equation (2) into:

$$\begin{eqnarray}&&n(\rho ,\lambda )=\sqrt{\displaystyle \frac{2{R}_{{\rm{S}}{\rm{M}}}(\lambda )\rho +1}{1-{R}_{{\rm{S}}{\rm{M}}}(\lambda )\rho }},\end{eqnarray} \tag{ 8 }$$

with R_SM(λ):

$$\begin{eqnarray}&&{R}_{\mathrm{SM}}{\left(\lambda \right)}^{2}=\displaystyle \frac{{D}_{1}{\lambda }^{2}}{{\lambda }^{2}-{C}_{1}}+\displaystyle \frac{{D}_{2}{\lambda }^{2}}{{\lambda }^{2}-{C}_{2}},\end{eqnarray} \tag{ 9 }$$

where C₁ and C₂ are the same absorption maxima as used in the fitting of Equation (7). Equation (9) is fit to the curves in Figure 5, yielding the best-fit parameters D₁ = 0.8416 and D₂ = 1.0592. We scaled the values of R_T(λ) to intersect R(632.8) = 0.2072 cm³ g⁻¹, as any deviation from this value is due to errors in the period determined in Section 3.1. Subsequently, Equation (9) and the Lorentz–Lorenz relation (Equation (2)) were used to calculate the n(λ) for different temperatures and the corresponding densities (as listed in Table 1). The resulting n(λ) curves are shown in Figure 6. These curves all show a similar wavelength behavior, with a nonlinear increase in n for shorter wavelengths. The curves differ with 0.018 for temperature steps of 20 K, resulting from the linear increase in the density as a function of temperature (Dohnálek et al. 2003). Table 2 summarizes a selected set of R_SM(λ) and n_T(λ) values for the investigated wavelength range. The full list with numbers is available from the supplementary information.

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Figure 6 shows the refractive index of ASW, with the corresponding densities in the wavelength range between 210 and 757 nm. Also shown is n(λ) of crystalline water ice at 150 K. The values of n(λ) are calculated from R_SM(λ) as derived in Section 3.4. A comparison of n(λ) from our crystalline ice deposited at 150 K with literature values of the refractive index as were reported by Warren & Brandt (2008) shows that these reproduce the literature values well. All of the values for n shown in this figure are available in Table 2.

With the experimental derivation of the real part of the refractive index, it is possible, in principle, to also derive the imaginary part, i.e., the parameter that describes the absorption properties, using the Kramer–Kronig relation. However, as already noted by Warren (1984), in the UV–vis, the imaginary refractive index is several orders of magnitude smaller than n, and as a direct result, in our experiments the accuracy of n is insufficient to constrain relevant imaginary refractive index values.