Rare gas mixtures

Using Eq. 7 it is now possible to calculate the matrix compressibility k. The observed linear pressure shift towards lower energies suggests that the interacting matrix atoms are located in the attractive part of the Lennard-Jones potential. The van der Waals potential is therefore expected to be a good approximation, which was employed in the derivation of the analytical expression for the pressure dependence (Eq. 8). Indeed, Tab. 4 shows that the results obtained, using the Lennard-Jones and the van der Waals potential respectively, are identical within the accuracy of our experiment. 

Of further interest is the question whether matrix correlations are of the same importance for the calculation of the matrix compressibility as they are for the potential parameter and the number density. A detailed analysis, however, reveals that for our model systems as well as for H2Pc-doped polymers PE and PS the corrections to the simple equations due to matrix correlations are smaller than the experimental error. This explains the good agreement between optically determined compressibilities, calculated conventionally without consideration of matrix correlations, and mechanically determined bulk values for polymeric systems. 

So far experiments with H2Pc-doped pure rare gas matrices have been reported. As discussed above, the transition from a polycrystalline to an amorphous solid can be studied in a single system using condensed rare gas mixtures, on which we will focus now. 

More insight into the dependence of the spectra upon the composition ratio can be gained if the frequency of the absorption maximum is plotted against the Xe concentration. This is done in Fig. 10.5, where the solvent shift is also givea The latter rises continuously 

The solid line in Fig. 10.5 was calculated with Eq. 1 extended to mixed systems. The density of the matrix units is interpolated from the values of the pure components assuming a constant atomic volume, as described in [39]. Using this simple approach, the calculated 

---

Rank D H, Rao B S and Wiggins T A 1963 Absorption speotra of hydrogen haiide-rare gas mixtures J. Chem. Phys. 37 2511-15... 

Henglein A (1956) The acceleration of chemical reactions of ultrasound in solutions of oxygen-rare gas mixtures. Naturwissenschaften (in German) 43(12) 277-278... 

Hope et al. (116) presented a combined volumetric sorption and theoretical study of the sorption of Kr in silicalite. The theoretical calculation was based on a potential model related to that of Sanders et al. (117), which includes electrostatic terms and a simple bond-bending formalism for the portion of the framework (120 atoms) that is allowed to relax during the simulations. In contrast to the potential developed by Sanders et al., these calculations employed hard, unpolarizable oxygen ions. Polarizability was, however, included in the description of the Kr atoms. Intermolecular potential terms accounting for the interaction of Kr atoms with the zeolite oxygen atoms were derived from fitting experimental results characterizing the interatomic potentials of rare gas mixtures. In contrast to the situation for hydrocarbons, there are few direct empirical data to aid parameterization, but the use of Ne-Kr potentials is reasonable, because Ne is isoelectronic with O2-. 

Subsequent work has revealed that collision-induced absorption is observable even in mixtures of monatomic gases, albeit not in unmixed monatomic gases. In rare gas mixtures, the translational absorption profile occurs in the microwave and far infrared regions. In mixtures of molecular gases, such translational absorption profiles are sometimes discernible but they are generally masked by the induced rotational bands mentioned. 

Interesting line narrowing has been observed of quadrupole-induced lines of hydrogen-rare gas mixtures. These have been explained by van Kranendonk and associates [428] in terms of the mutual diffusion coefficient of H2 in a rare-gas environment, as an effective lengthening of the interaction times of H2-atom complexes. 

Content. After a brief overview of molecular collisions and interactions, dipole radiation, and instrumentation (Chapter 2), we consider examples of measured collision-induced spectra, from the simplest systems (rare gas mixtures at low density) to the more complex molecular systems. Chapter 3 reviews the measurements. It is divided into three parts translational, rototranslational and rotovibrational induced spectra. Each of these considers the binary and ternary spectra, and van der Waals molecules we also take a brief look at the spectra of dense systems (liquids and solids). Once the experimental evidence is collected and understood in terms of simple models, a more theoretical approach is chosen for the discussion of induced dipole moments (Chapter 4) and the spectra (Chapters 5 and 6). Chapters 3 through 6 are the backbone of the book. Related topics, such as redistribution of radiation, electronic collision-induced absorption and emission, etc., and applications are considered in Chapter 7. 

The spectra shown in Fig. 3.1 appear as unstructured, broad absorption bands, with a maximum of absorption around 200 cm-1 for the lightest system and at lower frequencies for the more massive pairs. Absorption is weak, even at the peaks, and amounts to a mean absorption length of more than 1/a 106 cm (that is 10 km) if both gases are present at partial pressures of just one atmosphere. Absorption of rare gas mixtures increases, however, with increasing densities, with a mean absorption length of centimeters as we approach liquid densities. 

Molecular systems. Translational spectra like the ones shown for rare gas mixtures exist also for molecular gases and mixtures involving molecular gases. However, in that case, the rotational induced band will in general affect the appearance of the translational line since it appears generally at nearly the same frequencies. 

It is easy to see from Eq. 3.2 that for the rare gas mixtures, these equations are related to the moments M , according to... 

Table 3.1 lists measured spectral moments of rare gas mixtures at various temperatures. (We note that absorption in helium-neon mixtures has been measured recently [253]. This mixture absorbs very weakly so that pressures of 1500 bar had to be used. Under these conditions, one would expect significant many-body interactions the measurement almost certainly does not represent binary spectra.) For easy reference below, we note that the precision of the data quoted in the Table is not at all uniform. Accurate values of the moments require good absorption measurements over the whole translational frequency band, from zero to the highest frequencies where radiation is absorbed. Such data are, however, difficult to obtain. Good measurements of the absorption coefficient a(v) require ratios of transmitted to incident intensities, /(v)//o, that are significantly smaller than unity and, at the same time, of the order of unity, i.e., not too small. Since in the far infrared the lengths of absorption paths are limited to a few meters and gas densities are limited to obtain purely... 

Table 3.1. Binary integrated absorption coefficients of rare gas mixtures.

According to Eq. 3.5, yi may be considered the total absorption in the translational band. We, however, prefer to consider Mq the total intensity, Eq. 3.4 with n = 0, because the spectral function g(v) is more closely related to the emission (absorption) process than a(v). For rare gas mixtures, we have the relationships of Eqs. 3.7. In other words, yo may be considered a total intensity of the spectral function, g(v), and the ratio yi /yo is a mean width of the spectral function (in units of cm-1). Both moments increase with temperature as Table 3.1 shows. With increasing temperature closer encounters occur, which leads to increased induced dipole moments and thus greater intensities. 

Fig. 3.6. The spectral moment y as function of the product of densities, for various rare-gas mixtures at room temperature only one density was varied for each system the neon densities were fixed at 77, 31 and 46.5 amagats for the neon-argon, neon-krypton and neon-xenon mixtures, respectively and the krypton and xenon densities were fixed at 152 and 50 amagats, respectively, in their mixtures with argon. The departures from the straight lines seen at intermediate densities squared indicate the presence of many-body interactions. Reprinted with permission by Pergamon Press from [329].

Similar results were also obtained for argon-krypton mixtures [252]. Apart from the low-frequency region of the intercollisional dip, the variation of the translational line shape is rather subtle reduced absorption profiles of a number of rare gas mixtures at near-liquid densities (up to 750 amagat) have been proposed which ignore these variations totally [252],... 

A translational line like the one seen above in rare gas mixtures is relatively weak but discernible in pure hydrogen at low frequencies (<230 cm-1), Fig. 3.10. However, if a(v)/[l —exp (—hcv/kT)] is plotted instead of a(v), the line at zero frequency is prominent, Fig. 3.11 the 6o(l) line that corresponds to an orientational transition of ortho-H2. Other absorption lines are prominent, Fig. 3.10. Especially at low temperatures, strong but diffuse So(0) and So(l) lines appear near the rotational transition frequencies at 354 and 587 cm-1, respectively. These rotational transitions of H2 are, of course, well known from Raman studies and correspond to J = 0 -> 2 and J = 1 — 3 transitions J designates the rotational quantum number. These transitions are infrared inactive in the isolated molecule. At higher temperatures, rotational lines So(J) with J > 1 are also discernible these may be seen more clearly in mixtures of hydrogen with the heavier rare gases, see for example Fig. 3.14 below. 

The binary spectra of hydrogen-helium mixtures, Fig. 3.12, differ from the spectra of pure hydrogen, Fig. 3.10, especially by the translational line (familiar from the spectra of rare gas mixtures) whose intensity increases strongly with increasing temperature. Moreover, the rotational line intensities when normalized by the product of helium and hydrogen... 

Isotope spectra. Rotovibrational spectra of deuterium, and of deuterium-rare gas mixtures, have also been recorded over a wide range of temperatures and densities [342]. The differences between the H2-X and D2-X spectra (with X = H2 or D2, respectively, or a rare gas atom) are much like what has been seen above for the rototranslational spectra. 

Furthermore, sometimes a much less pronounced absorption dip is seen at the rotovibrational transition frequencies. Knowledge of the dip is nearly as old as collision-induced absorption itself the earliest report [129] mentions an unexplained component X at about 4100 cm-1, observed in hydrogen-rare gas mixtures. Subsequent studies [120, 121, 175] pointed out the main features of the new phenomenon. Specifically, it was noted that... 

Later studies showed the same phenomena in deuterium and deuterium-rare gas mixtures [335, 338, 305], and also in nitrogen and nitrogen-helium mixtures [336] in nitrogen-argon mixtures the feature is, however, not well developed. The intercollisional dip (as the feature is now commonly called) in the rototranslational spectra was identified many years later see Fig. 3.5 and related discussions. The phenomenon was explained by van Kranendonk [404] as a many-body process, in terms of the correlations of induced dipoles in consecutive collisions. In other words, at low densities, the dipole autocorrelation function has a significant negative tail of a characteristic decay time equal to the mean time between collisions see the theoretical developments in Chapter 5 for details. 

In an attempt to model the spectral functions of rare gas mixtures, Fig. 3.2, it was noted that a Gaussian function with exponential tails approximates the measurements reasonably well [75], about as well as the Lorentzian core with exponential tails. Two free parameters were chosen such that at the mending point a continuous function and a continuous derivative resulted the negative frequency wing was again chosen as that same curve, multiplied by the Boltzmann factor, to satisfy Eq. 3.18. Subsequent work retained the combination of a Lorentzian with an exponential wing and made use of a desymmetrization function [320],... 

Method of moments. In rare gas mixtures, the induced dipole consists of just one B component, with Ai AL = 0001, Eq. 4.14. Alternatively, one particular B(c) component may cause the overwhelming part of a measured spectrum, like the quadrupole-induced component in mixtures of small amounts of H2 in highly polarizable rare gases ((c) = Ai AL = 2023, Eq. 4.59) in a given spectral range, other components (like 0001, 2021,...) are often relatively insignificant. In such cases, one can write down more or less discriminating relationships between certain spectral moments of low order n that are obtainable from measurements of the collision-induced spectral profile, g Al(o>),... 

The simplest systems of interest here are the rare gas mixtures. It had been argued that the translational spectra of the rare gas pairs, when expressed in terms of reduced frequencies, v = v/vmax, and reduced intensities, a = a/amax, all look roughly the same [22] the principle of... 

In recent years, a dependable dipole function for He-Ar, last column of Table 4.3, has been obtained [278] which we compare with the universal dipole function mentioned [23], Fig. 4.5. The He-Ar interaction potential is one of the better known functions [13] and suggests Rmj = 6.518 bohr. Both functions were normalized to unity at the separation R = 5 bohr in the figure. The comparison shows that at small separations the logarithmic slope of the most dependable dipole function is roughly one half that of the universal p, and p diverges rapidly from p(R) for R — o. Similar discrepancies have been noted for other rare gas systems (Ne-Ar, Ne-Kr, and Ar-Kr [152]). Even if for these other systems the dipole function is not as well known as it is for He-Ar, it seems safe to say that for the rare gas mixtures mentioned the induced dipole function is definitely not identical with the universal function at the distances characteristic of the spectroscopic interactions the universal dipole function is not consistent with some well established facts and data. We note that the ratio of // (/ ) and the He-Ar potential is indeed reasonably constant over the range of separations considered (not shown in the figure). 

In recent molecular dynamics studies attempts were made to reproduce the shapes of the intercollisional dip from reliable pair dipole models and pair potentials [301], The shape and relative amplitude of the intercollisional dip are known to depend sensitively on the details of the intermolecular interactions, and especially on the dipole function. For a number of very dense ( 1000 amagat) rare gas mixtures spectral profiles were obtained by molecular dynamics simulation that differed significantly from the observed dips. In particular, the computed amplitudes were never of sufficient magnitude. This fact is considered compelling evidence for the presence of irreducible many-body effects, presumably mainly of the induced dipole function. 

Theoretical estimates of these were given for some rare gas mixtures elsewhere, based on the one-effective electron model of Jansen [173], The conclusion was reached that triatomic dispersion dipoles must be added to the near-range induced dipoles just mentioned because of strong cancellations of the near- and distant-range components [172],... 

In this Chapter, we consider the theory of collision-induced absorption by rare gas mixtures. We look at various theoretical efforts and compare theoretical predictions and computations with measured spectra and other experimental facts. The theory of induced absorption is based on quantum mechanics, but in certain cases, the use of classical physics may be justified, or indeed be the only viable choice. The emphasis will be on the computation of induced absorption by non-reactive, small atomic systems in the infrared. Diatomic and triatomic systems show most of the features of collisional absorption without requiring complex theory for their treatment. The theory of induced absorption of small clusters involving molecules will be considered in Chapter 6. 

For the rare gas mixtures, i) and /) represent the initial and final translational states. The integration is over positive frequencies so that only the terms corresponding to E, < Ef survive,... 

For rare gas mixtures, hcvft — Ef — E, is the energy difference between final and initial translational states. The limitation of the integration to... 

Comparison with measurement. Measurements of the absorption of rare gas mixtures exist for some time. This fact has stimulated a good deal of theoretical research. A number of ab initio computations of the induced dipole moment of He-Ar are known, including an advanced treatment which accounts for configuration interaction to a high degree see Chapter 4 for details. Figure 5.5 shows the spectral density profile computed... 

Constant acceleration approximation. An approximation introduced to the time-dependent intermolecular correlation function G, which was commonly referred to as the constant acceleration approximation (CAA), was used to compute the line shapes of collision-induced absorption spectra of rare gas mixtures, but the computed profiles were found to be unsatisfactory [286], It does not give the correct first spectral moment. 

It is, therefore, interesting to point out that in a recent molecular dynamics study, shapes of intercollisional dips of collision-induced absorption were obtained. These line shapes are considered a particularly sensitive probe of intermolecular interactions [301]. Using recent pair potentials and empirical pair dipole functions, for certain rare-gas mixtures spectral profiles were obtained that differ significantly from what is observed... 

---