1. Introduction
Interest in spectroscopic investigations of the different forms of ice mainly lies in the fact that ice is a typical hydrogen-bonded crystal, the simplest one, and some of its crystalline forms are orientationally disordered.
The first spectra of ice Ih single crystals obtained with modern Raman laser techniques were those of Faure (unpublished) and Faure and Chosson ([ c 1971]) for the translational region, and these spectra already give a good idea of the general shape of the spectrum and of some of its details. More detailed spectra, but with polycrystalline samples, were taken by Wong and others (1973) and Wong and Whalley (1976).
The first aim of this paper will be to obtain a better knowledge of the details of the spectrum of the single crystal using polarized light. We then propose to account for the existence of translational frequencies up to c. 320 cm-1 by TO–LO splittings in connection with which we discuss briefly a dynamical model already presented (Faure and Chosson, [ c 1971]; Faure and Kahane, [c1971]).
Fig. 1. Unpolarized Raman spectra of single-crystal ice Ih at 180 K.
Fig. 2. Polarized Raman spectra of single-crystal ice Ih at 180 K.
In ice Ih the oxygen atoms are arranged in D 6h 4 symmetry, with four oxygen atoms in the unit cell. It is generally accepted that proton positions are governed only by the well-known Bernai–Fowler rules (1933) and that these proton positions are therefore disordered. The corresponding cubic form, ice Ic, is also generally considered to be disordered.
II. Experimental Methods and Spectra
The Raman spectra were recorded in the range 15 to 350 cm-1 with a PH1 Coderg double monochromator and a Spectra-Physics 164 argon-ion laser (4880 and 5 145 Å; typically 1 W) from pure transparent samples of H2O ice Ih single crystals, cut in the form of small parallelepipeds or cylinders according to the experiments planned. The recording temperatures were 77 and 180 K, with polarized and unpolarized light, spectral slit widths used varied from a to 0.5 cm-1 and scanning speeds from 25 to 1 cm-1/min (Figs 1–3). The spectra exhibit much fine structure but not as much as in Wong and others (1973) for polycrystalline samples. The reproducible features are listed in Table I together with the corresponding features from Wong and others (1973). The Y(ZZ) X polarization is very similar to the spectrum with unpolarized light, and the differences between the four polarizations are not as great as is generally observed in ordered crystals. Nevertheless, depending on the polarization, one can notice, in addition to the different intensities, slight frequency shifts in the maxima near 225 cm-1, larger shifts (c. 8 cm-1) in the maxima near 300 cm-1 (as previously observed by Faure and Chosson ([e1971])) and differing shapes for the maxima near 60 cm-1 and for the shoulders near 270 cm-1 (Table I). These results show the interest of studying single crystals of ice Ih in polarized light, in spite of the proton disorder. In particular the observation of the two maxima near 300 cm-1 (307 and 298.8 cm-1 at 180 K) is not possible in unpolarized light owing to the width of the peaks, and we believe the existence of these two maxima is important for the proposed interpretation of the spectrum.
Fig. 3. X1 (1 Y) Z1 polarization for different θ = (q, c) at 77 K.
Table I. Frequencies in cm-1 of the Features in the Raman Spectrum of ice Ih (H2O)
III. Activity of Order-Allowed and Disorder-Allowed Modes
The D 6h symmetry of the oxygen-atom lattice allows three modes A 1g , E lg , and E 2g for first-order ordinary Raman scattering processes. Clearly however the spectra do not consist of three well-defined lines. The temperature dependence of intensity (Faure and Chosson, [c1971]) shows that the c. 300, c. 225, and c. 60 cm-1 maxima correspond to one-phonon processes. In fact the Raman spectrum is very similar to the frequency spectrum derived from slow-neutron scattering experiments (Prask and others, 1968). Furthermore, first-order infrared processes are all forbidden in the D 6h symmetry of the oxygen atoms of ice Ih ; however the translational infrared spectrum (Bertie and others, 1969) exists and is also very similar to this same derived frequency spectrum. The same vibrations seem to occur in both infrared and Raman spectra but with different relative intensities. (This simultaneous activity will direct us in the choice of the proposed interpretation.) Whalley and Bertie (1967) have given an explanation for these similarities. The disorder of the protons gives rise to what they call disorder-allowed modes in Raman and infrared first-order processes, not only for q ? o (q being the phonon wave vector) but also for q in the whole Brillouin Zone. The Raman spectrum is then due to these disorder-allowed modes together with the order-allowed A 1g , E 1g , and E 2g modes in the D 6h symmetry.
A satisfactory description of the observed infrared and Raman intensities requires not only the knowledge of the frequencies at Γ (centre of the Brillouin Zone) but also of the dispersion curves and the frequency spectrum.
The dynamical models already used (Kahane, unpublished; Faure, 1969; Wong and others, 1973; Shawyer and Dean, 1972; Prask and others, 1972; Renker, 1973; Bosi and others, 1973) either are unable to account for the existence of translatory fundamentals up to c. 320 cm-1, or use numerous force constants (ten for instance) and are therefore of doubtful physical significance. All these models use only short-range valence forces.
IV. To-Lo Splittings in Disordered Ice—A Simple Dynamical Model
In ordered crystals, phonons active in first-order infrared absorption produce an electric dipole moment in the lattice, and the accompanying long-range electric fields lead to splittings between transverse optic and longitudinal optic modes (Poulet, unpublished; Loudon, 1964; Poulet and Matthieu, [ c 1970]). If these phonons are simultaneously active in the first-order Raman effect (a necessary condition being that the crystal lacks an inversion centre) the TO and LO modes can be observed by the Raman effect under appropriate experimental conditions (sec for instance Arguello and others (1969)).
As we have seen in Section III, in ice Ih and Ic where the proton positions are nearly disordered, all the translational vibrations are simultaneously Raman and infrared active (at least weakly) and in particular those for q ? 0. If these vibrations are sufficiently delocalizcd, one can expect TO–LO splittings in the Raman spectrum. In particular the strong infrared absorption near 225 cm-1 will generate a TO–LO splitting and the Raman peaks near 225 and 300 cm-1 will be due to this splitting, since the Lyddane–Sachs–Teller relation ω LO/ω TO = (ε0/ε∞)[inline-1] applied to these maxima leads to a value of c. 1.3 for ω LO/ω TO as against c. 1.23 for (ε0/ε∞)[inline-1].
We thus explain semi-quantitatively the existence of translational fundamentals beyond c. 250 cm-1 by TO–LO splittings. To be more quantitative one needs a dynamical model leading to TO–LO splittings. The simplest way of removing the inversion centre of the D 6h symmetry and reaching the desired aim in dynamical calculations is to attribute to the oxygen atoms positive and negative effective charges, with opposite charges for two neighbouring oxygen atoms. The D 6h symmetry lowers to C 6v and the different modes transform as indicated in Table II. A 1 and E l are “polar” modes. In the (K, G, ρ) dynamical model we consider here, K is the stretching constant for an O—O bond, G is the angle bending constant for an O—O—O angle and ρ (ρ ? q 2/4πε0 a 3) accounts for the long-range interaction of the effective charges (q being the effective charge and a the distance between two first-neighbour oxygen atoms). In this first step we have only three parameters.
Some indications concerning the calculation of the frequencies at the centre of the Brillouin Zone are given by Faure and Kahane ([ c 1971]). These frequencies expressed in K, G, and ρ are given in Table II for q||c and q ╧ c.
The looseness of the selection and polarization rules on account of the proton disorder does not facilitate an unambiguous choice for the numerical values of K, G, and ρ. As our aim was to test the ability of this mixed Coulomb–valence-dynamical model to reproduce the high-frequency part of the translational spectrum, we chose the simple fit A 1 (LO) = 299 cm-1, E 1(TO) = 225 cm-1, and B 2 = 170 cm-1 (Table II). As well as accounting for the maxima near 300, 225, and 170 cm-1, this model leads to a possible explanation for the frequency shifts already described as a function of polarization of the maxima near 300 and 225 cm-1 since the two calculated LO modes have slightly different frequencies, as have the two calculated TO modes.
Table II. Calculated Frequencies at Γ with (K, G, ρ) Model for Ice Ih
We do not find any frequency at Γ near 60 cm-1, but for E 2 we find c. 92 cm-1. With only three parameters K, G, and ρ it is not possible to lower this E 2 mode down to c. 60 cm-1 without destroying the good fit obtained for the A l and E 1 modes. On introducing two different angle bending constants G and G′ this becomes possible. With this (K, G, G′, ρ) model we obtain the same numerical values as in Table II for A 1(LO), A 1(TO), E 1(LO), E 1(TO) and the two B 2 modes, but E 2 = 217.6 cm-1 and E 2 = 57 cm-1 for K = 21.34, G = 8.034, G′ = —7.050 and ρ = 5.163 N m-1 (details will be published later).
With the (K, G, ρ) model, the dispersion curves and the frequency spectrum were calculated and compared with the corresponding measurements (Prask and others, 1968; Renker, 1973; unpublished work by Renker and Parisot). They will be published elsewhere. The merely moderate agreement obtained for the very-low-frequency part of the translational spectrum (as expected, since E 2 is found at c. 92 cm-1 and the calculated elastic constants are rather high) would be very likely to be improved by the use of the (K, G, G, ρ) model.
V. Directional Effects in Ice
The theory for polar modes in non-cubic crystals predicts the existence of “quasi-longitudinal” and “quasi-transverse” modes, the frequencies of which vary with the angle θ between the phonon wave vector q and the optical c-axis. This was investigated in ice by rotating a single crystal in the form of a cylinder inside the low-temperature cell so as to obtain different values of θ. In order to get maximum sensitivity, these polarized spectra were recorded in the 300 cm-1 region, since the observed frequency shifts near 300 cm-1 are greater than those near 225 cm-1. At 77 K the ? 1(? 1 Y)? 1 polarization frequency varies from 309.5 cm-1 for θ ? o to 317 for θ ? 90° (Fig. 3). At the same temperature for another sample with the arrangement of Figure 2 the two different measured frequencies of the polarizations, which we ascribe to LO modes on the basis of the results of the (K, G, ρ) model, are 309.6 cm-1 and 316.6 cm-1 (Table I). The important fact is that this θ dependence occurs (within the experimental error due to the width of the peaks) inside the interval between the two LO modes.
If this directional effect in ice Ih is indeed q-dependent (experiments are being undertaken to confirm this point), it would be an argument for the proposed explanation of the existence of translational frequencies up to c. 320 cm-1 in the Raman spectrum of Ice Ih, since q-dependent directional effects are characteristic of uniaxial crystals possessing TO–LO splittings.
Acknowledgements
We are indebted to Dr J. Klinger for growing the single crystals and to Mr Merlin for the computer programming.
Discussion
W. B. KAMB: TO what extent is the explanation of TO–LO splitting given by your effective-charge model basically the same as, or different from, the explanation given by Klug and Whalley in the previous paper?
P. FAURE: We propose to explain the existence of translational frequencies up to about 320 cm-1 in the Raman spectra of ice Ih and Ic by the TO–LO splittings necessary from the observed infrared absorption. We believe that this explanation is basically the same as the one given by Klug and Whalley. In order to generate TO–LO splittings in the dynamical calculation in a simple manner, we use positive and negative permanent charges and this is the difference with Klug and Whalley’s more elaborate model.
We had already used this explanation involving TO–LO splittings for ice in a previous paper but expressed in a different way.
E. WHALLEY: The relation between the models of Faure and Chosson and of Klug and Whalley can be summarized as follows. Both attribute the scattering and absorption at c. 300 cm-1 to LO modes. Faure and Chosson interpret this in molecular terms by assuming that alternate water molecules carry a positive and negative charge, whereas Klug and Whalley assume that the molecules are uncharged but each has an effective charge tensor which describes the dipole moments induced by translations. In ordered cubic ice the charge tensors of nearest-neighbour molecules have the same magnitude but opposite sign. In disordered ice, the charge tensors presumably vary from molecule to molecule in an irregular manner depending on the orientation of the molecule and its neighbours.
