Q branch

Figure Bl.4.9. Top rotation-tunnelling hyperfine structure in one of the flipping inodes of (020)3 near 3 THz. The small splittings seen in the Q-branch transitions are induced by the bound-free hydrogen atom tiiimelling by the water monomers. Bottom the low-frequency torsional mode structure of the water duner spectrum, includmg a detailed comparison of theoretical calculations of the dynamics with those observed experimentally [ ]. The symbols next to the arrows depict the parallel (A k= 0) versus perpendicular (A = 1) nature of the selection rules in the pseudorotation manifold.

Beeause AL = 0 transitions are allowed for n vibrations, one says that n vibrations possess Q- branches in addition to their R- and P- branehes (with AL = 1 and -1, respeetively). 

The absorption that is "missing" from the figure below lying slightly below 2900 em is the Q-branch transition for whieh E = E it is absent beeause the seleetion rules forbid it. 

Figure 6.27 The IJSq, 77 — infrared band of acetylene. (The unusual vertical scale allows both the very intense Q branch and the weak P and R branches to be shown conveniently)...

The separation of individual lines within the Q branch is small, causing the branch to stand out as more intense than the rest of the band. This appearance is typical of all Q branches in infrared spectra because of the similarity of the rotational constants in the upper and lower states of the transition. 

The effective value of B, for the lower components of the doubled levels, can be obtained from the P and R branches by the same method of combination differences used for a type of band and, for the upper components, from the Q branch. From these two quantities and may be calculated. 

The effect of the AK = 1 selection rule, compared with AK = 0 for an transition, is to spread out the sets of P, Q, and R branches with different values of K. Each Q branch consists, as usual, of closely spaced lines, so as to appear almost line-like, and the separation between adjacent Q branches is approximately 2 A — B ). Figure 6.29 shows such an example, E — A band of the prolate symmetric rotor silyl fluoride (SiH3F) where Vg is the e rocking vibration of the SiH3 group. The Q branches dominate this fairly low resolution specttum, those with AK = - -1 and —1 being on the high and low wavenumber sides, respectively. 

The selection rules are the same for oblate symmetric rotors, and parallel bands appear similar to those of a prolate symmetric rotor. However, perpendicular bands of an oblate symmetric rotor show Q branches with AK = - -1 and — 1 on the low and high wavenumber sides, respectively, since the spacing, 2 C — B ), is negative. 

The method of combination differences applied to the P and R branches gives the lower state rotational constants B", or B" and D", just as in a A transition, from Equation (6.29) or Equation (6.32). These branches also give rotational constants B, or B and D, relating to the upper components of the 77 state, from Equation (6.30) or Equation (6.33). The constants B, or B and D, relating to the lower components of the state, may be obtained from the Q branch. The value of q can be obtained from B and B. ... 

Fig. 0.2. (a) The comb spectrum of N2 considered as a quantum rotator. The envelope of the rotational structure of the Q-branch slightly split by the rotovibra-tional interaction is shaded, (b) The depolarized rotovibrational spectrum of N2 at corpuscular density n = 92 amagat, T = 296 K and pressure p = 100 atm. The central peak, reported in a reduced (x30) scale is due to a polarized component [5] (V) experimental (—) best fit. 

Chapter 3 is devoted to pressure transformation of the unresolved isotropic Raman scattering spectrum which consists of a single Q-branch much narrower than other branches (shaded in Fig. 0.2(a)). Therefore rotational collapse of the Q-branch is accomplished much earlier than that of the IR spectrum as a whole (e.g. in the gas phase). Attention is concentrated on the isotropic Q-branch of N2, which is significantly narrowed before the broadening produced by weak vibrational dephasing becomes dominant. It is remarkable that isotropic Q-branch collapse is indifferent to orientational relaxation. It is affected solely by rotational energy relaxation. This is an exceptional case of pure frequency modulation similar to the Dicke effect in atomic spectroscopy [13]. The only difference is that the frequency in the Q-branch is quadratic in J whereas in the Doppler contour it is linear in translational velocity v. Consequently the rotational frequency modulation is not Gaussian but is still Markovian and therefore subject to the impact theory. The Keilson-... 

Storer model used in this theory enables us to describe classically the spectral collapse of the Q-branch for any strength of collisions. The theory generates the canonical relation between the width of the Raman spectrum and the rate of rotational relaxation measured by NMR or acoustic methods. At medium pressures the impact theory overlaps with the non-model perturbation theory which extends the relation to the region where the binary approximation is invalid. The employment of this relation has become a routine procedure which puts in order numerous experimental data from different methods. At low densities it permits us to estimate, roughly, the strength of collisions. 

Fig. 0.4. Experimental nitrogen Q-branch of coherent anti-Stokes Raman scattering spectrum (CARS) measured at 700 K and different pressures [14].

The envelope of the Stark structure of the rotator in a constant orienting field, calculated quantum-mechanically in [17], roughly reproduces the shape of the triplet (Fig. 0.5(c)). The appearance of the Q-branch in the linear rotator spectrum indicates that the axis is partially fixed, i.e. some molecules perform librations of small amplitude around the field. Only molecules with high enough rotational energy overcome the barrier created by the field. They rotate with the frequencies observed in the... 

The origin of the rotational structure of the isotropic Q-branch (Av = 0, Aj = 0) is connected with the dependence of the vibrational transition frequency shift on rotational quantum number j [121, 126]... 

In order to describe the shape of the Q-branch after its collapse, it is sufficient to use the stochastic perturbation theory expounded in the... 

Owing to this fact, they may be sewed together in order to describe quantitatively the Q-branch transformation with density from a gas to the liquid state [133-5]. 

Fig. 3.2. Q-branch transformation with increase of density in strong collision (a) and weak collision (b) approximation at T = 0.1 (I) T = 0.3 (II) T = 10 (III). All spectra are normalized to 1 at their maxima.

The quasi-classical description of the Q-branch becomes valid as soon as its rotational structure is washed out. There is no doubt that at this point its contour is close to a static one, and, consequently, asymmetric to a large extent. It is also established [136] that after narrowing of the contour its shape in the liquid is Lorentzian even in the far wings where the intensity is four orders less than in the centre (see Fig. 3.3). In this case it is more convenient to compare observed contours with calculated ones by their characteristic parameters. These are the half width at half height Aa)i/2 and the shift of the spectrum maximum ftW—< > = 5a>+A, which is usually assumed to be a sum of the rotational shift 5larger scale A determined by vibrational dephasing. 

Fig. 3.4. The isotropic Q-branch width (a) and rotational shift (b) calculated in the models of strong (1) and weak (2) collisions as a function of r E = ojqte and T = 1/tj correspondingly. The straight lines are perturbation theory estimates of spectral width and shift...

Fig. 3.5. The shift of oxygen Q-branch [137]. The contribution of vibrational dephasing A that is linear in density is shown as a straight line. The initial deviation of this line reproduces the theoretical behaviour of 5(n) shown in Fig. 4.4(b). The experiment was performed at a few temperatures higher than the critical one 7.87 K (A), 0.95 K (o) and 0.12 K ( ).

Fig. 3.8. The Q-branch Raman width alteration with condensation of nitrogen. The theoretical results for the strong (A) and weak (B) collision limits are shown together with experimental data for gaseous [89] ( ) and liquid nitrogen [145] ( ) (point a is taken from the CARS experiment of [136]). The broken curves in the inset are A and B limits whereas the intermediate solid curve presents the rotational contribution to line width at y = 0.3. The straight line estimates the contribution of vibrational dephasing [143], and the circles around it are the same liquid data but without rotational contribution.

Fig. 3.9. (a) Dependence of the experimental half-width of the isotropic Q-branch of N2 and CO on the density ( ) CO, 295 K (+) CO in CF4, 273 K (A) CO in C02> 323 K (O) N2> 295 K (A) N2 in C02> 323 K. The error in the measurements of half-width is +0.2 cm-1, (b) The same data as in (a) but in relation to measured T = I/ojqXj. Theoretical curves for strong (curve 1) and weak (curve 2) collision limits are identical to curves A and B in Fig. 3.8. The upside-down triangles near the broken line present the difference between the actual half-width of CO in CO2 (A) and curve 1. The error in all cases is approximately the same as that indicated for CO.

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

The Q-branch band shape in the Keilson-Stoier model... 

Fig. 3.11. Comparison of side branch broadening with Q-branch nonbroadening , made for nitrogen in [160] for 27°C and different pressures 15 atm (curve a), 25 atm (curve b), 40 atm (curve c), 60 atm (curve d). In the lower part 8a> is the width of resolved rotational components, 5v is the width of the non-resolved Q-branch, which is primarily isotropic.

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