A new technique for determining the refractive index of ices at cryogenic temperatures.

A reflection-absorption optical (RAO) spectrometer, operating across the ultra-violet/visible (UV/visible) wavelength region, has been developed that allows simultaneous measurements of optical properties and thickness of thin solid films at cryogenic temperatures in ultrahigh vacuum. The RAO spectrometer enables such measurements to be made after ice deposition, as opposed to most current approaches which make measurements during deposition. This allows changes in the optical properties and in the thickness of the film to be determined subsequent to thermal, photon or charged particle processing. This is not possible with current techniques. A data analysis method is presented that allows the wavelength dependent n and k values for ices to be extracted from the reflection-absorption spectra. The validity of this analysis method is shown using model data from the literature. New data are presented for the reflection UV/visible spectra of amorphous and crystalline single component ices of benzene, methyl formate and water adsorbed on a graphite surface. These data show that, for benzene and methyl formate, the crystalline ice has a larger refractive index than amorphous ice, reflecting changes in the electronic environment occurring in the ice during crystallisation. For water, the refractive index does not vary with ice phase.


Introduction
In order to understand the nature of materials in remote, extreme or production environments, observations of radiation scattering have long been employed. Encoded within the scattered radiation are details of size, shape and spectroscopy (composition) of the scattering centres. These can be interpreted using appropriate theoretical descriptions for bulk materials, thin films or granular materials. Knowledge of the optical constants of materials is fundamental to interpreting this data across the electromagnetic spectrum.
However, it is especially important in the infrared and optical regions when considering dust grains and thin films growing on them in our atmosphere, throughout the solar system and in the interstellar medium [1][2][3][4][5] . Likewise, measurements of optical constants play an important characterisation and quality assurance role in a range of fields from geology 6 to the food industry 7 and optical technologies. [8][9][10] Of particular relevance to this work is the examination of optical constants of astrochemical ice analogues. [11][12][13] Ices form on dust grains in space at low temperatures when atoms reactively accrete to form molecules and gas phase molecules accrete onto dust grains in a range of astrophysical environments. 14 The optical constants define how light interacts with a material. For a non-absorbing medium, the key optical constant is the refractive index, n, which is given by the ratio of the velocity of light in a vacuum to the propagation velocity of light in the material. However, this description does not take into account the absorption of light by the material. To do this, we must define the complex refractive index, m, which is given in terms of refraction and absorption coefficients, n and k respectively (equation 1).
(1) m = n + ik As both n and k are strongly dependent on wavelength, m itself must also be wavelength dependent. The value of m, through the Fresnel coefficients for reflection and transmission 15,16 , determines the ratio of reflected and transmitted intensities as light propagates across an interface between two different media.
Typically, the Kramers-Kronig (KK) relationship has been used to determine ice refractive indices from experimental spectra. 11,13,[17][18][19][20][21][22][23][24][25] The KK method is a powerful tool, but several issues have recently been identified that can result in the recovery of incorrect or incomplete refractive index data. 26 In particular, the KK method requires an accurate value of the ice thickness, d. This is often estimated using assumptions 21,[27][28][29][30] or only measured during ice deposition. 11,22,23,[31][32][33][34][35] He-Ne laser interferometry is often used to make such measurements. Figure 1 illustrates the production of interference fringes as light is reflected from a reflective substrate supporting a thin transmissive ice film. Measuring the thickness of an ice during deposition means that the effect of post-deposition processing or particle bombardment 36 on its thickness cannot be measured and therefore neither can the refractive index. 26 Finally, in order to perform the KK analysis the Fresnel coefficients for transmission are required.
Therefore, only ices deposited on a transmitting substrate can be examined and experimental arrangements utilising the significant sensitivity advantage of reflection-absorption on a reflective substrate are not suitable. represent the incident and reflected beams respectively. θ 0 , θ 1 and θ 2 are the angles of incidence, refraction due to the ice and refraction due to the HOPG substrate respectively. The ice thickness is d, and n 1 and k 1 are the ice optical parameters. n 0 is the refractive index of the vacuum, equal to 1, and n 2 and k 2 are the substrate optical properties.
In light of the above, we have designed and developed a novel RAO spectroscopy apparatus to simultaneously determine the UV/visible refractive index and thickness of ices deposited on a reflective surface under ultrahigh vacuum. This apparatus has previously been demonstrated in measurements of the real part of the refractive index of amorphous benzene (C 6 H 6 ) ice as a proof of concept. 26 The apparatus allows n and d to be simultaneously determined ab initio and in a single experimental procedure. Additionally, as spectra are recorded after the ice has been deposited, the effect of processing can be measured. 26 The work described here presents a comparison of the refractive index of amorphous and crystalline C 6 H 6 , methyl formate (CH 3 OCHO, MF) and water (H 2 O) ices and provides an explanation for the differences observed. These systems are being investigated in this work due to their relevance to astrochemistry [37][38][39][40][41]  transition were sharper and shifted to higher energy loss in crystalline benzene compared to amorphous ice. 50 In addition to examining the phase change itself, several authors have found that C 6 H 6 deposited at elevated temperatures directly forms an ordered or crystalline phase. When C 6 H 6 is deposited at 110 K on Cu(111), an initial chemisorbed layer forms, lying flat on the surface. 51 Subsequently deposited C 6 H 6 adsorbs as an almost parallel oriented physisorbed bilayer, which is followed by the growth of ordered multilayers. 51 This behaviour was shown by a combination of temperature programmed desorption (TPD), high resolution EELS (HREELS) and near edge X-ray absorption fine structure (NEXAFS) measurements. 51 Similar results were also observed for C 6 H 6 deposited at 120 K on Ru(001), shown by TPD and HREELS. 51 On graphite, neutron diffraction experiments suggest that an initial, flat-lying, physisorbed monolayer forms when C 6 H 6 is deposited at 100 K, followed by the growth of crystalline multilayers. 52 The effect of the benzene phase change on the electronic spectrum of C 6 H 6 ice was recently examined in detail by Dawes et al. 45 Transmission spectra were recorded in the wavelength range 115 -340 nm for C 6 H 6 deposited on a MgF 2 window at 25 K. C 6 H 6 was crystallised by annealing the ice deposited at 25 K to 90 K for 2 minutes, and subsequently several band shifts and line profile changes were observed in the spectrum.
Whilst the C 6 H 6 phase change has previously been described spectroscopically, the effect of the phase change on the refractive index of C 6 H 6 ices has not been examined. Indeed only one study has determined the refractive index of crystalline C 6 H 6 ice. Romanescu et al. split the output of a HeNe laser into two beams which were then reflected off a Au substrate at different reflection angles during deposition of C 6 H 6 at 100 K. 54 The period of the resulting interference fringes, and the known reflection angles, were used to determine an n value of 1.54 ± 0.02 for the crystalline C 6 H 6 ice. 54 This is higher than the value of n = 1.43 ± 0.07 previously determined by our group for amorphous C 6 H 6 ice. 26 In addition to comparing the real parts of the refractive indices of amorphous and crystalline

Experimental Methodology
The data presented here were recorded using a novel reflection absorption optical spectrometer, 26 attached to an ultra-high vacuum (UHV) chamber. 56 A highly oriented pyrolytic graphite (HOPG) surface is mounted in the vacuum chamber on the end of a copper cold finger, connected to a closed-cycle helium refrigerator (SHI-APD). This is used as a carbonaceous interstellar dust grain analogue surface. 12,[57][58][59][60][61][62] The helium refrigerator allows a base temperature of 26 K to be achieved and the typical base pressure in the chamber is ≤ 2 × 10 -10 mbar. The sample is cleaned by heating to 250 K for 3 minutes, and cleanness is confirmed by the lack of any desorption, as measured by a mass spectrometer, during a heating ramp.
The design and installation of the RAO apparatus has been described in detail elsewhere. 26 Briefly, a UHV compatible, stainless steel lens assembly was designed which allowed UV/visible light to be fed into the chamber, reflected from the HOPG substrate and subsequently collected via fibre optic cables and analysed by a compact dispersive spectrophotometer (Ocean Optics QE Pro). This assembly is connected to a linear drive (Vacgen Ltd.) which allows the angle of incidence (and hence angle of reflection) to be varied from near normal to almost grazing incidence. The range of angles of incidence achieved is  is not equal across the entire spectral range. 63 All UV/visible spectra presented are the average of 128 scans with an integration time of 250 ms per scan. These parameters were chosen by trial and error to obtain the highest signal without saturating the CCD array detector. RAIR spectra for ices were also recorded using a Thermo-Nicolet FTIR spectrometer coupled to a liquid nitrogen cooled mercury cadmium telluride detector. RAIR spectra are the result of the co-addition of 256 scans at a resolution of 4 cm -1 and are also presented in the form of ΔR/R.
Where ices have been annealed, the ices were deposited at base temperature and subsequently heated linearly to the annealing temperature. They were held at the annealing temperature for 3 minutes before returning to base temperature and recording a UV/visible or RAIR spectrum.

Data analysis
The UV/visible spectra were first analysed using a method developed by Harrick for use in the infrared 64 , and adapted to our system in previous work. 26 This allows the ice thickness, d, and the real part of the refractive index, n, to be determined by examining the spacing between the extrema of the interference fringes produced at different angles of incidence. Unlike previous methods, n and d are determined using a single experimental procedure, and any effect of processing the ice on these values can be examined as they are measured after the ice has been grown. Whilst this serves as a proof of concept for the apparatus and gives accurate values of n and d for the ices examined, this method does not give information about the imaginary part of the refractive index, k, nor the wavelength dependence of the complex refractive index. This is essential if the data are to be used in simulating the optical properties of ices in astronomical environments.
In order to determine m as a function of wavelength for an optically opaque surface, a method was developed based on the work of Kozlova et al. 55 The analysis works by taking a starting value of n, along with other parameters described below including an initial guess of k, and using these to simulate a reflectance spectrum, (ΔR/R) sim . The simulated spectrum is then compared to the experimental one, (ΔR/R) exp , and a match parameter is determined. The values of n and k are then varied until a good match between (ΔR/R) exp and (ΔR/R) sim is found.
With reference to Figure 1, the input parameters can be determined. The angle of incidence, θ 0 , refractive index of the vacuum, n 0 (equal to 1), complex refractive index of the ice, m 1 (where m 1 = n 1 + ik 1 ), ice thickness, d, and the complex refractive index of the HOPG, m 2 (equal to n 2 + ik 2 ) all contribute to (ΔR/R) exp . In the case of HOPG, there are two sets of optical parameters as HOPG is birefringent: one set for p-polarised light (n 2p and k 2p ) and the second for s-polarised light (n 2s and k 2s ) yielding complex refractive indices m 2p and m 2s . Both of these optical parameters must be taken into account in the analysis, as the incident light is unpolarised. In this work, the optical parameters used for HOPG are those determined by Djurišić and Li. 65 Using the parameters described, the Fresnel coefficients of reflection for s-and p-polarised light for the vacuum/ice, ice/substrate and vacuum/substrate interfaces, can be determined.
These are r 1p and r 1s (vacuum/ice), r 2p and r 2s (ice/substrate) and r 0,2p and r 0,2s (vacuum/substrate) as given by equations (2) -(7). 16,55 (2) In equations (2) -(7), θ 1 , θ 2p and θ 2s are the complex angles of refraction at each interface given by equations (8)  Using the Fresnel coefficients, it is possible to calculate R sim and R 0sim , which are comprised of the reflection coefficients for s-and p-polarised light, R s and R p respectively, for the ice, and R 0s and R 0p for the clean surface. The reflection coefficients are calculated using equations (11) - (14). 55 (11) R 0s = | r 0,2s | 2 In equations (11) and (12), δ is the complex phase thickness and describes the phase change introduced in the reflected light. It is given by equation (15) 55 where λ is the wavelength of the light. R 0sim and R sim are given by equations (16) and (17). 55 (16) R sim = R p + R s (17) R 0sim = R 0p + R 0s Using equations (2) -(17), a simulated reflectance spectrum, (ΔR/R) sim , for a given ice can be produced. This can be compared to the experimental spectrum, (ΔR/R) exp , to give a match parameter, s, which is calculated according to equation (18).
The parameter s is minimised using a least squares function and, via an iterative process, the target parameters, n 1 and k 1 , can be determined.
A Python code was written to perform the analysis described. The required inputs are the experimental reflection spectra at each reflection angle and the ice thickness examined, with the reflection angles and thicknesses determined using the Harrick method. 26,64 Additionally, the HOPG optical parameters are necessary. The outputs of the analysis code are the In order to test the theory and the code, test data with known values of n 1 and k 1 were used as a benchmark. The values of n 1 were those of Kofman et al. 66 determined for amorphous solid water (ASW). Figure 2 plots n as a function of wavelength determined by Kofman et al. 66 The data are fitted with a polynomial function in order to determine n across the full wavelength range used in this work. It should be noted that the wavelength range has an upper limit of 591 nm due to the availability of the HOPG optical parameters.  66 The blue line is a fit to the data to give n across the wavelength range used to test the analysis method.
Kofman et al. 66 did not determine wavelength dependent values of k for ASW, therefore a constant value of 0.01 was assumed. This value was chosen, as previously reported values in the relevant wavelength region tend to be small (<< 0.5) and positive for water. 25,67 In any case, the specific value used is less important than the fact that it is known. The values of n and k for ASW were used to produce simulated reflection spectra using a separate Python code, shown in the ESI Appendix E2. The code takes the known values of θ, d, λ, n 2s , k 2s , n 2p , k 2p , n 1 and k 1 and uses equations (2) - (17) to give simulated ΔR/R spectra. Spectra were simulated with ice thicknesses between 100 and 500 nm. An example of a simulated water spectrum is shown in Figure 3A (blue trace) for an ASW ice thickness of 400 nm and a reflection angle of 53°.
The simulated ASW spectra were then used as input (ΔR/R) exp data for the analysis code.  Figure 3B shows that the calculated n values from the simulated ASW spectra agree very well with those determined by Kofman et al. 66 Similarly, Figure 3C shows that the determined k values are in excellent agreement with the assumed value of 0.01, with only very small variations observed around 350 and 560 nm. In fact, the determined values of k differ from the assumed value by less than 0.05%. Figure 3A shows that this small variation is insignificant, as the simulated spectrum matches the input spectrum very well. Clearly the data analysis method presented here works well for the analysis of UV/visible reflectionabsorption spectra of ices on opaque or reflective surfaces.

Experimental Results
Initially, RAIRS was used to characterise the ices grown on HOPG. Infrared spectra of C 6 Figure 4 shows several RAIR spectra of C 6 H 6 ices. Figure 4A shows the wavenumber region 3150 -2950 cm -1 and Figure 4B shows the region 1600 cm -1 -950 cm -1 . The red spectrum shown in Figure 4 is that of 500 L m of C 6 H 6 deposited on HOPG at 27 K. At this deposition temperature, benzene is in its amorphous form. 44,45 The bands in the amorphous C 6 H 6 spectrum at 3090 cm -1 and 3032 cm -1 ( Figure 4A) and those at 1479 cm -1 and 1036 cm -1 ( Figure 4B) have been observed previously. 42 They are assigned to the C-H stretch (ν(CH) aromatic ), ν(CH) combination mode, aromatic C-C stretch (ν(CC) aromatic ) and C-H in plane deformation (δ(CH) in-plane ) mode of C 6 H 6 respectively 68,69,79 . In addition to these bands, several other features are observed in the spectra, which can be assigned by comparison with the literature. 48,52,68,69,79 The band at 3068 cm -1 is assigned to the ν(CH) aromatic mode of C 6 H 6 . The bands at 1176 cm -1 and 1148 cm -1 are assigned to δ(CH) in-plane modes and the band at 973 cm -1 is the δ(CH) out of plane mode. Tentative assignments of the C 6 H 6 ice is known to undergo a thermally induced phase change. 44,45,47,50,51,80 This phase change has previously been observed upon annealing to 90 K, 44 with total crystallisation being complete following annealing to 120 K. The blue trace in Figure 4 shows the effect of annealing the amorphous C 6 H 6 ice to 90 K. In Figure 4A, the ν(CH) aromatic mode at 3090 cm -1 splits into two bands at 3091 cm -1 and 3085 cm -1 , the second ν(CH) aromatic mode red-shifts to 3066 cm -1 and the ν(CH) combination band splits into two bands at 3037 cm -1 and 3030 cm -1 . In Figure   4B, are in agreement with those previously assigned to crystallisation of the ice. 44,45,48 Further experiments were undertaken to examine whether crystalline C 6 H 6 ice could be grown directly. A deposition temperature of 70 K was chosen as it was sufficiently high to crystallise C 6 H 6 , but far enough away from the desorption temperature of around 150 K, as shown in Figure E1, ESI. The green trace in Figures 4A and 4B shows the RAIR spectrum of 500 L m of benzene deposited at 70 K. It is clearly identical to the annealed spectrum, and upon further annealing no changes were observed in any of the bands, indicating that the ice was fully crystalline. In light of these results, crystalline C 6 H 6 ices were grown at a deposition temperature of 70 K, and amorphous C 6 H 6 ices were deposited at base temperature.
Similar characterisation experiments were performed for MF and H 2 O ices, both of which also undergo thermally induced phase changes. RAIR spectra of MF are shown in Figure E2 in the ESI. Additionally, band assignments are given in Table E1  in an attempt to grow crystalline ice directly, the RAIR spectrum of the ice differs to that of the annealed ice ( Figure E2, ESI) and agrees with the spectra recorded by Modica and Palumbo 70 . Previous studies have shown that MF can exist in two crystalline forms 82 , in line with our observations. In this work, only the form deposited at 105 K is examined, and this is referred to as crystalline MF. Figure E3 of the ESI shows RAIR spectra of H 2 O ices grown at 27 K and at 135 K. At 27 K, the broad bands are typical of ASW, whereas the spectrum resulting from the ice grown at 135 K is typical of that observed for crystalline water ice (CI) 14,84 . Figure 5 shows UV/visible spectra of amorphous ( Figure 5A) and crystalline ( Figure 5B) C 6 H 6 ices for a single C 6 H 6 exposure and at two reflection angles. Several common features are immediately apparent. Firstly, the feature in all the spectra at 656 nm arises as an artefact from the light source and can be disregarded. 26,63 Interference fringes are also observed.
These are used to determine the ice thickness and real part of the refractive index using the method of Harrick. 26,64 The extrema of the interference fringes shift in wavelength at different reflection angles and previous work has shown that increasing the exposure increases the number of interference fringes seen in the spectrum, as expected due to the increased ice thickness. 26 In each spectrum in Figure 5, absorption bands, which do not shift with C 6 H 6 exposure, are also observed. The amorphous C 6 H 6 spectra ( Figure 5A) show two absorption features. One is located at 214.8 nm and the other is observed as several peaks centred on approximately 255 nm. These absorption bands have been observed previously [45][46][47]50,[85][86][87] and are assigned to the 1 B 1u  1 A 1g and 1 B 2u  1 A 1g transitions of C 6 H 6 , respectively. The 1 B 1u  1 A 1g band in the amorphous C 6 H 6 spectra in Figure 5A appears as a broad peak with a maximum intensity at 214.8 nm. In the crystalline spectra ( Figure 5B), the intensity of this feature is lost. The reason for the disappearance of the 1 B 1u  1 A 1g band upon crystallisation is unclear. However, it should be noted that the intensity of the reflected light below approximately 220 nm is very low in our experiment, which may affect the clarity of the spectra below this wavelength.  Figure 6 shows UV/visible spectra of amorphous and crystalline C 6 H 6 in the region 230 nm -280 nm, to highlight the 1 B 2u  1 A 1g transition. It is clear from Figure 6 that the absorption  Dawes et al. 43 The positions of the peaks in amorphous and crystalline C 6 H 6 ice are shown in Table 1, alongside the observations of Dawes et al. 45 The spectrum of crystalline C 6 H 6 ice in Figure 6 also contains minor peaks in between the bands listed in Table 1. These are possibly evidence of Davydov splitting, 88 where the presence of more than one equivalent molecular entity in the unit cell interacts to cause the splitting of electronic or vibrational bands. This effect has also been observed in UV/visible spectra of crystalline to determine the real part of the refractive index and the thicknesses of the ices. The determined vales of n are shown in Table 2. The refractive index of amorphous C 6 H 6 ice has previously been determined for the data presented here and has a value of n = 1.43 ± 0.07. 26 The data shown in Figure 5 give a refractive index of 1.63 ± 0.08 for crystalline C 6 H 6 ice. This is in agreement, within error, with the value of n = 1.54 ± 0.02 determined by Romanescu et al. 54  The determined real parts of the refractive indices In Table 2 were then used to calculate the thicknesses of the ices. Figure 7 shows that for all amorphous and crystalline ices, there is a linear relationship between thickness and exposure. Errors in d are determined from the upper and lower limits of n.
In all cases, it is clear that crystalline ices are thinner than the amorphous ones. Tests were performed to ensure that the observed thickness difference was due to the phase change, rather than to a reduced sticking probability during deposition at elevated temperatures.
Temperature programmed desorption, where the surface is linearly heated whilst desorption is monitored by a quadrupole mass spectrometer (QMS) was used to confirm this. The area under a plot of QMS intensity as a function of surface temperature, for a given mass fragment, is directly proportional to the amount of adsorbate on the surface 14  small decrease was found, which was corrected for by increasing the dose time at the higher deposition temperature. Hence, the change in thickness is solely a product of the ice phase.
The thickness difference between amorphous and crystalline C 6 H 6 ices can be expressed as a compaction factor, calculated from the ratio of weighted least squares linear fits to each set of data in Figure 7. In each fit, the intercept is fixed at 0, as an exposure of 0 L m must have no thickness. The ratio of the gradients for the crystalline to amorphous C 6 H 6 data gives a compaction factor of 0.74. This means that crystalline C 6 H 6 ice is 0.74 times as thick as the amorphous ice for the same total amount of C 6 H 6 . The compaction factors for MF and H 2 O are 0.74 and 0.82 respectively.
Following the initial analysis, UV/visible spectra of amorphous and crystalline C 6 H 6 ices were then used as input spectra for the analysis code, in order to determine the wavelength dependent values of n and k for the ices. Figure 8 shows the output spectrum (blue trace) for an exposure of 400 L m C 6 H 6 at a fixed reflection angle of 53° in the wavelength region 200 -300 nm. The 1 B 1u  1 A 1g and 1 B 2u  1 A 1g transitions at 214 nm and 255 nm respectively are very well reproduced in the simulated spectrum. The values of n and k over the same wavelength range for C 6 H 6 are shown in Figure 9. The good agreement between the spectra in Figure 8, as well as the data already shown in Figure 3, indicates that the theory presented here is suitable to describe the physical processes which give rise to the observed reflection spectra.  Figure 10 shows the same spectra as Figure 8, but now across the entire wavelength range.
Whilst the data shown in Figure 8 show that the absorption features of benzene can be well reproduced by the code, Figure 10 shows that the interference fringes at longer wavelengths are not well reproduced by the code. Discussion For both C 6 H 6 and MF, Table 2 shows that the crystalline ices have a higher refractive index.
This suggests that light impinging on the solid from a vacuum is slowed more upon entering the crystalline ice medium than the amorphous one. Similar behaviour has been reported for then films of germanium selenide. 94 This change can be rationalised by considering the concept of the optical impedance, Z, of a medium. As proposed by Kronig 95 , optical impedance is defined as the ratio of the amplitudes of the perpendicular wave vectors in a propagating light wave, i.e. the ratio of the electronic wave vector, E, to the magnetic wave vector, H, and is shown in equation (19). 95,96 Also shown in equation (19) is the relationship between impedance and the permeability, µ, and dielectric constant, ε, of a medium. 96 In the case of non-magnetic media, such as C 6 H 6 or MF, µ = 1 and hence Z = ε -1/2 . 96 In an amorphous form, the molecules are randomly aligned, and hence the electronic environment is disordered, i.e. the electron cloud of each molecule is randomly aligned relative to its neighbours. Crystalline C 6 H 6 ice consists of C 6 H 6 molecules arranged in a herringbone structure. 48,52,[97][98][99][100] Therefore, there is a difference in the electronic environment in the two ice phases for each molecule. This will cause the dielectric constant, and hence impedance, of the ice to change. In order to explain the difference in refractive index between amorphous and crystalline ice, it is necessary to look at the relationship between the refractive index n and the optical impedance Z. The literature shows that n varies with ε 1/2 . 16,95 Therefore n ∝ 1/Z, and hence a change in impedance must be accompanied by a change in refractive index.
For C 6 H 6 and MF, the refractive index of crystalline ice is higher than that of amorphous ice; 1.63 ± 0.08 compared to 1.43 ± 0.07, and 1.23 ± 0.07 compared to 1.40 ± 0.08 respectively.
This implies that Z is lower for crystalline C 6 H 6 and MF ices than for the amorphous phase, and subsequently that the dielectric constant, ε, of the crystalline ice is higher than for amorphous ice, as shown in equation (19). In agreement with this, crystalline materials have previously been shown to have higher values of ε than their amorphous counterparts. 82 theoretical work by Pilling and Bonfim also showed that an increase of ε would lead to an increase in the intensity of infrared bands, in agreement with the spectra shown in Figure 4 and Figure E2. 103 The dielectric constant of a medium is also related to its polarizability, 103  investigations of this type on other molecules will help to clarify this.
It was shown that the amount of adsorbate on the HOPG surface was consistent for all molecules at the various deposition temperatures. Therefore, the observed compaction shown in Figure 7 is solely a consequence of the ice phase transition. This implies that crystalline ices are denser than their amorphous counterparts. However, care must be taken to not over-extrapolate from this data. The exact structures of amorphous C 6 H 6 and MF ices are unknown. For example, if the ices are porous, the compaction may be due to a combination of pore collapse and decrease in intermolecular distances. For H 2 O ice, the structure is well studied 35,84,93,104,105 and ASW is known to be porous. In this work, the ASW is likely to be in its compact form, where the porosity is less than other forms of ASW but still significant 104,105 . Despite the fact that the refractive index of ASW and CI does not vary (Table   2), there is a clear compaction shown in Figure 7. This is certainly in part due to loss of porosity.
Westley et al. 35 found that in the temperature range 20 K -140 K, the density of H 2 O ice remains constant. In this case, loss of porosity going from ASW to CI would be the sole reason for the observed compaction. Based on the compaction factor of 0.82, this suggests that porosity represents 18% of the ASW thickness, higher than typical estimates of 10%.
Therefore, for H 2 O ice, we conclude that a loss of porosity and a decrease in intermolecular distance upon the phase change is responsible for the observed compaction. Indeed, an increase in density of H 2 O ice as a function of surface temperature has been previously observed 93,106 .
The analysis used to determine the complex refractive index of an ice was shown to be valid using test data from the literature, as shown in Figure 3. However, when the code is used to analyse the experimental C 6 H 6 spectra, there is clearly a discrepancy between the input data and results ( Figure 10). This suggests that there is a systematic error in the experimental data which is not accounted for in the detailed analysis. The analysis reproduces the C 6 H 6 absorption features (Figure 8), but the interference fringes at wavelengths greater than approximately 270 nm are not well reproduced ( Figure 10). It is noted that the absorption features do not shift as the reflection angle is varied, whereas the relative positions of the interference fringes do shift, as shown in Figure 5. It is therefore possible that an experimental uncertainty on the true value of θ leads to the observed discrepancy in Figure 10. This uncertainty is likely due to the limits of precision of the differentially pumped rotary feedthrough which enables sample rotation, and the need to rotate the sample between recording a background spectrum and a sample spectrum. An alternative reason for the disagreement shown in Figure 10 is that the structure of amorphous C 6 H 6 ice is unknown. The presence of pores or small nuclei would cause a large amount of scatter and diffuse reflections of the light which would impact on the spectra, converting them from reflection-absorption spectra to diffuse-reflectance spectra. However, a similar discrepancy between the experimental and simulated spectra also exists for crystalline C 6 H 6 ice. Therefore, we suggest that an error in reflection angle is a more likely explanation for this situation.
In order to test the effect of a mismatch between the experimental reflection angle and that used in the analysis code, the benchmarking procedure already described was repeated, but with an error of ± 2° built into the value of θ. It was shown that this relatively small error on the experimental reflection angle, compared to the value of θ used in the analysis, has a dramatic effect on the results of the analysis, leading to inaccurate values of n and k for the ice (see Figure E6, in the ESI). Furthermore, it results in a poor match between the interference fringes of the experimental and simulated spectra, similar to that shown in Figure 10 for the experimental data ( Figure E7 in the ESI). Therefore we conclude that an uncertainty on reflection angle is the most likely cause of the discrepancy between the data sets shown in Figure 10. This uncertainty was not found to impact the Harrick analysis 26,64 , as this method is dependent on the fringe spacing rather than the exact fringe positions. Experimental modifications are currently underway to remove this systematic error from the experiments.
Nonetheless, the data presented here clearly demonstrate the usefulness of this new technique for simultaneously determining ice thickness and refractive index, as well as the applicability of the analysis method to the recorded data.

Conclusions
We have presented a novel technique to simultaneously measure the refractive index and thickness of ices adsorbed on opaque substrates at cryogenic temperatures and UHV pressures. New data are reported for amorphous and crystalline C 6 H 6 , MF and H 2 O ices adsorbed on an HOPG substrate. RAIRS shows that these phases can be identified from their infrared absorption spectra, as reported previously. UV/visible reflection-absorption spectra have allowed the determination of the refractive indices of the distinct ice phases.
Crystalline C 6 H 6 is found to have a refractive index of 1.63 ± 0.08, whilst amorphous C 6 H 6 ice has a lower refractive index of 1.43 ± 0.07. 26 Similarly, crystalline MF ice has a refractive index of 1.40 ± 0.08 compared to 1.23 ± 0.05 for amorphous MF. This difference is assigned to a change in the dielectric constant of the ice that occurs as a result of the phase change. H 2 O shows different behaviour, with the two phases having almost identical refractive indices, in line with previous work. Further work is required to fully understand this behaviour.
We describe a novel analysis method to determine the wavelength dependent, complex refractive index of these ices. This method was tested using data from Kofman et al. 66 and was shown to correctly determine the complex refractive index of an ice from a UV/visible reflection-absorption spectrum. Applying the method to our experimental data has shown that the absorption features of amorphous benzene ice are excellently reproduced by the analysis method. This work has implications for the astrochemistry community, where the complex refractive index is used to simulate the optical properties of icy grains to account for scattering effects in spectroscopic measurements.
The method we have described is widely applicable to the measurement of refractive indices and thicknesses of ices in situ, thus allowing measurements to be made of ices that have been processed by heating, photon or charge particle irradiation. This provides a distinct advantage over previous techniques, 11