Quadrupole induced

As if can be seen in fhese simulations, all MAS spectra are shifted towards low frequencies wifh respecf fo fhe true isotropic chemical shift—an effect known as the second-order quadrupole-induced shift. The centre of gravity ( g) of fhese lineshapes is given by ° ... 

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. 

A rotation of the H2 molecule through 180° creates an identical electric field. In other words, for every full rotation of a H2 molecule, the dipole induced in the collisional partner X oscillates twice through the full cycle. Quadrupole induced lines occur, therefore, at twice the (classical) rotation frequencies, or with selection rules J — J + 2, like rotational Raman lines of linear molecules. Orientational transitions (J — J AM 0) occur at zero frequency and make up the translational line. Besides multipole induction of the lowest-order multipole moments consistent with... 

H2-X where X is a molecule. If a molecule other than H2 is chosen as the collision al partner X, new absorption bands appear at the rotovi-brational bands of that molecule. As an example, Fig. 3.17 shows the rototranslational enhancement spectra [46] of H2-CH4 for the temperature of 195 K. At the higher frequencies (v > 250 cm-1), these look much like the H2-Ar spectrum of Fig. 3.10 the H2 So(J) lines at 354, 587, and 815 cm-1 are clearly discernible. Besides these H2 rotational lines, a strong low-frequency spectrum is apparent which corresponds to the (unresolved) induced rotational transitions of the CH4 molecule these in turn look like the envelope of the rotational spectra seen in pure methane, Fig. 3.22. This is evident in the decomposition of the spectrum, Fig. 3.17, into its main components [46] the CH4 octopole (dashed curve) and hex-adecapole (dot-dashed curve) components that resemble the CH4-CH4 spectrum of Fig. 3.22, and the H2 quadrupole-induced component (dotted curve) which resembles the H2-Ar spectrum, Fig. 3.14. The superposition (heavy curve) models the measurement (big dots) closely. Similar spectra are known for systems like H2-N2 [58]. 

Fig. 3.17. The rototranslational spectrum of H2-CH4 at 195 K experimental points big dots H2 quadrupole-induced component dotted CH4 octopole-induced component dashed CH4 hexadecapole-induced component dot-dashed total heavy curve. Reproduced with permission from the National Research Council of Canada from [46].

Fig. 3.33. Analysis of the fundamental band of normal hydrogen at 20.4 K into its 11 main components. Overlap-induced components Q(0) and Q( 1) (widely spaced dashes) are broader than the quadrupole-induced components (closely spaced dashes, from left to right Qq(1), Si(0) and Si(1)) and double transitions (dotted Qq(0) and Qq( 1) ei(l)+S0(0) and Q,(0) + Sb(0) e,(l) + So(l) and 2i(0) + So(1)- The dots represent the summation of these the measurement is shown as a heavy line. Reproduced with permission from the National Research Council of Canada from [414],...

Fig. 3.48. Enhancement of the absorption in the fundamental band in a hydrogen-argon mixture at low and high densities at 152 K the profiles are normalized to give Si(l) the same peak intensity. The argon density was 8 ama-gat (solid line) and 820 amagat (dashed line), respectively. The density splitting of the (overlap-induced) Q branch and the density narrowing of the S lines are apparent. A new (quadrupole-induced) Q line appears inthe wide absorption dip (between Qp and Qr) observed at high density. Reproduced with permission from the National Research Council of Canada from [137].

From the beginnings, attempts to model the line shapes of collision-induced absorption spectra were based on the assumption that the various rotational lines of induced spectra, Figs. 3.10 through 3.14, are superpositions of scaled and shifted line profiles, g(C)(v), °f a small number of different, e.g., overlap- and quadrupole-induced, types [313, 404],... 

If at least one of the interacting particles is a molecule, further induction mechanisms arise. Molecules are surrounded by an electric field which may be viewed as a superposition of multipole fields. A collisional partner will be polarized in the multipole field and thus give rise to induced dipole components. In the case of symmetric diatoms like H2 or N2, the lowest-order multipole is a quadrupole and asymptotically, for R - 00, the quadrupole-induced dipole may be written as [288, 289]... 

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>),... 

H2 quadrupole moment, <72(re) at the fixed equilibrium position, and thus the long-range coefficient of the quadrupole-induced dipole component, Eq. 4.3, is about 5% too small relative to the proper vibrational average, <12 = (v = 0 < 2(r) f = 0) [216, 217, 209], A 5% difference of the dipole moment amounts to a 10% difference of the associated spectral intensities. Furthermore, the effects of electron correlation on this long-range coefficient can be estimated. Correlation increases the He polarizability by 5% but decreases the H2 quadrupole moment by 8% [275], a net change of-3% of the leading induction term B R). 

Table 4.12 shows that several of these correction coefficients are of nearly the same magnitude but of opposite sign, for example, for the = 2023 and 0223 components, (00 yl 01) and (00, 4 0T). Therefore, in Fig. 4.3 we need to give only one representative example of these functions. A similar situation exists for the four correction terms of the 2021,0221 components. The comparison of Figs. 4.2 (right-hand plot) and 4.3 shows that the correction terms belonging to the same /l AL set show very much the same dependence as the dipole coefficient for the rotationless case, i.e., the quadrupole induced terms reflect the R 4 dependence... 

Related to this near absence of logarithmic curvature of the wings is the fact that the KO model is superior in describing line profiles resulting from overlap induction. The BC shape, on the other hand, shows more curvature in the wing, as this is needed for the modeling of profiles generated by low-order multipolar induction. Purely quadrupole-induced components are closely modeled by the BC shape. 

Fig. 5.8. Root mean square relative errors of model line shapes fitted to a quantum profile, the quadrupole-induced (XL = 23) component [69], The abscissa gives the ratio of peak intensity and wing intensity of the fitted portion of the exact profile. The superiority of the BC model (lower set of data points) over the desymmetrized Lorentzian (upper set) is evident.

Criteria for choosing specific models. We have seen above that spectral profiles of binary complexes can be computed from a rigorous quantum formalism. These describe the measurements well. Long profiles, i.e., profiles with a peak-to-wing intensity of several orders of magnitude, can be readily obtained. As an illustration of how well the various available model profiles approximate the exact computations, we show results for one basic profile type characteristic of absorption by H2-H2 pairs [69], a pure quadrupole-induced profile which accounts for roughly 90% of the total intensity of the rototranslational H2-H2 collision-induced absorption spectra at 77 K. 

The BC and K0 models, on the other hand, show much smaller root mean square errors, typically in the 1 % range, over an amazingly substantial range x of intensities fitted, Fig. 5.8, lower set of data points. Maximal deviations from the exact profiles amount to no more than twice the root mean square errors shown, that is well within the experimental uncertainties of the best measurements. The BC model is especially well suited to approximate quadrupole-induced profiles. The K0 model, on the other... 

Spectral moments can also be computed from classical expressions with Wigner-Kirkwood quantum corrections [177, 189, 317] of the order lV(H2). For the quadrupole-induced 0223 and 2023 components of H2-H2, at the temperature of 40 K, such results differ from the exact zeroth, first and second moments by -10%, -10%, and +30% respectively. For the leading overlap-induced 0221 and 2021 components, we get similarly +14%, +12%, and -56%. These numbers illustrate the significance of a quantum treatment of the hydrogen pair at low temperatures. At room temperature, the semiclassical and quantum moments of low order differ by a few percent at most. Quantum calculations of higher-order moments differ, however, more strongly from their classical counterparts. 

Hj-Ar-Ar rototranslational band. Experimental studies of the density variation of the rototranslational collision-induced absorption spectra of argon gas with a small admixture of hydrogen or deuterium have been reported [140, 108, 109, 106], Since there is no induced dipole component associated with Ar-Ar interactions, the spectroscopically dominant three-body interactions involve one hydrogen molecule and two argon atoms, H2-Ar-Ar. These spectra consist mainly of the quadrupole-induced rotational So(J) lines arising from the XL = 23 component. 

The quadrupole-induced components A1A2AL = 0223,2023 are the most important components of the spectrum. Figure 6.3 compares the positive-frequency wings of the various spectral functions, Eq. 6.55, at the temperature of 77 K. These consist of free — free and bound — free components, the first and third terms to the right of Eq. 6.55. The dimer structures were suppressed in Fig. 6.3 but are shown (as obtained in the isotropic interaction approximation) in Fig. 6.4. The low-frequency end of the profiles, Fig. 6.3, may be considered a low-resolution rendition (as may be obtained with a monochromator of low resolution, 10 cm-1 pressure broadening would similarly flatten the spectral dimer structures). 

Where in Fig. 6.3 two curves are drawn with the same type of line, the upper curve describes the free — free transitions of the collisional pair, and the lower one the bound — free contributions. Asymptotically, at high frequencies, the bound — free contributions amount to only about 1% of the free — free components. At the lower frequencies, however, the bound — free components are relatively more significant. In fact, the bound — free components which must be superimposed with the free — free components to obtain the spectral function, Eq. 6.54, affect the shapes of the profiles near the line centers. We note that the quadrupole-induced components 0223, 2023 do not feature a bound — bound spectrum, but the less important overlap components 0221, 2021 do. However, absorption due to this overlap component is insignificant, a few percent of the total absorption. 

Fig. 6.5. Computed structures due to the hydrogen dimer, in the quadrupole-induced (0223,2023) components near the So(0) line center at 120 K (the temperature of Jupiter s upper atmosphere). Superimposed with the smooth free — free continuum (dashes) are structures arising from bound — free (below 354 cm-1) and free - bound (above 354 cm-1) transitions of the hydrogen pair (dotted). The convolution of the spectrum with a 4.3 cm-1 slit function (approximating the instrumental profile of the Voyager infrared spectrometer) is also shown (heavy line) [282].

The computed profiles are shown in Figs. 6.11 and 6.12. The various components labeled XL = 01, 21, 23, and 45 are sketched lightly. Their sum is given by the heavy curve marked total. The spectra consist of a broad, purely translational part that is dominated at the low frequencies by the isotropic component (XL = 01). Other, generally smaller contributions are noticable, the most significant of which is the quadrupole-induced component (XL = 23) which shapes the rotational, induced lines, So(J) with J = 0,1,..., of H2 this component arises from a dipole component... 

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