Transition double

Apparently, the direct transition from vapor to solid is less common than the double transition vapor — liquid — solid, see, e.g., Refs.158-160). From the rate of solidification of metal droplets (average diameter near 0.005 cm) at temperatures 60° to 370° below their normal melting points, the 7sl was concluded158) to be, for instance, 24 for mercury, 54 for tin, and 177 erg/cm2 for copper. For this calculation it was necessary to assume that each crystal nucleus was a perfect sphere embedded in the melt droplet the improbability of this model was emphasized above. 

It is worth noting that a full pH range scan for an equimolar proportion of acid and amine, i.e., a horizontal cut in the middle of the map, would result in a similar double transition, with an extremely wide central region with WII phase behavior and precipitated catanionic species, with two Will bands at high and low pH, and eventually a small WI region at extreme high and low pH values. 

Fig. 3.22. Rototranslational absorption spectra of CH4-CH4 pairs [141] at 296 K in the frequency range 50-400 cm-1. A simple decomposition of the measurement ( ) is attempted in terms of an octopole-induced component (a) a hexadecapole-induced component (b) and double transitions (c) the superposition (heavy) reproduces the measurement closely.

Fig. 3.32. Rotovibrational term scheme of H2 showing single and double transitions of pairs after [187],...

Specifically, the eleven profiles include the obvious single transitions, i.e., the rotovibrational transitions in just one of the two colliding H2 molecules these are the Si(0), Si(l), and Qi (1) transitions in one of the two interacting molecules. Double transitions in both collisional partners are also taking place, such as the simultaneous transitions gi(l)+So(0) (which occur near the Si(0) transition frequency) and Q (1) + So(l) (near Si(l)), Fig. 3. 32. Intensities of all these lines are known from theory (classical multipole approximation, Chapter 6) their superposition reproduces the measurement closely, Fig. 3.33. 

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],...

Overtone bands. The induced first overtone band of H2 is shown in Fig. 3.37 at a variety of temperatures, observed in pure hydrogen gas using long absorption paths. Instead of the three components Q, S(0) and S(l) seen in the fundamental band, we now observe much richer structures, especially at the lower temperatures. This fact suggests that a number of double transitions take place. If one constructs a rotovibrational term scheme of the H2 molecule, like Fig. 3.32, which includes the lowest rotational levels of the v = 2 vibrational state, this is obvious. Various... 

Double transitions. In molecular fluids, simultaneous or double transitions occur at sums and differences of the rotational lines, with the absorption of a single photon. Several of such double transitions have been pointed out above, Figs. 3.32 and 3.37. In general terms, one may say that these occur at sums and differences of the rotovibrational (and/or electronic) transition frequencies of the molecules involved, as was explained in the discussions related to Fig. 1.3. 

Early attempts have modeled the positive frequency wing (v > 0) with the help of a Lorentzian, Eq. 3.15, because of a perceived similarity of the observed induced lines with the Lorentzian no theoretical justification was pretended. The negative frequency wing may then be described by exp (—hcv/kT) times that Lorentzian so that Eq. 3.18 is satisfied [215, 188, 414, 411]. Systematic deviations from this model were, however, noticed in the wings [75, 244]. Nevertheless, beautiful analyses of various rotational and rotovibrational induced bands were thus possible and significant new knowledge concerning the role of overlap and multipole induction, double transitions, etc., was obtained in this way [422],... 

The most striking fact about the dipole matrix elements for the H2-H2 overtone band is the large number of components, Tables 4.13 and 4.14. Besides the single vibrational transitions (V2 = 0 — 2 while t i = const, Table 4.13, and vice versa), we now have to consider vibrational double transitions ( i = 0 — 1 and V2 —> 1, Table 4.14). The associated spectra appear at nearly the same frequencies. If one adds to these the various rotational bands, Eq. 4.40, a very large number of spectral components arises that must be accounted for in the computations of the overtone... 

It is well known that in n-body complexes rotational transitions of the order n may occur [400]. However, we will assume here that the interaction forces are pairwise additive and isotropic so that rotations and translations are uncorrelated. In this case, at most double transitions occur [400] and the correlation function of the total dipole moment can be written as... 

H2-H2 overtone band. Figure 6.10 shows the results of similar calculations for the first overtone band of H2 (heavy line) [284]. Two measurements (dots and circles) are also given for comparison. The light dotted and dashed lines give intermediate results, corresponding to single and double transitions the latter are even more numerous in the overtone band than they were in the fundamental band. The comparison of such calculations with measurements at a variety of temperatures shows satisfactory agreement [284]. 

Herzberg was able to point out another line at 816.6 nm which he identified as a double transition, partially overlapped by an adjacent CH4 band in the Uranus spectra. In the laboratory spectra recorded with unmixed hydrogen, this double transition was relatively strong, but in the photographic plates of Uranus the feature was much weaker relative to the S3(0) line. This observation led Herzberg to conclude that sizeable He concentrations exist in these atmospheres (albeit the estimates of [He] [H2] abundance ratio seem high), because the S3(0) feature is enhanced by the presence of He, but H2-He pairs cannot undergo double transitions these features thus appear weak in Kuiper s plates relative to the S3(0) feature. 

It is well known that the tetrahedral frame of the CH4 molecule is easily distorted. If the tetrahedral frame of CH4 were robust, the purely rotational infrared spectra of CH4 would not exist. However, even at temperatures as low as room temperature, the CH4 molecule features hundreds of very weak, dipole-allowed rotovibrational lines at frequencies from 42 to 208 cm-1, the so-called groundstate to groundstate (gs—>gs) transitions. Moreover, more than 1500 weak, dipole-allowed transitions exist within the polyad system v /v — 1/2/1, at frequencies from 14 to 500 cm-1 [42]. These allowed transitions arise from distortions of the tetrahedral frame by rotation and the internal dynamics of the CH4 molecule, due to the coupling of normal modes of the flexible CH4 frame. Collisional frame distortion should probably be associated with unresolved gs— gs and similar polyad bands. Some evidence of such collision-induced bands of CH4 in CH4-X complexes has been pointed out [39-41]. Besides these collision-induced bands that presumably are due to collisional frame distortion of CH4, fairly significant, unexplained collision-induced bands also exist that are shaped by rotovibrational transitions of the collisional partner X = H2, N2, or CH4, and by double transitions of the bimolecular CH4-X complex [39-41]. 

In the broad region labeled 2, there is a set of concentric ovals labeled with the lowest powers of 2. These mark period doubling transitions whose repetitions lead to chaos. If we calculate the response to the forcing of a small departures from this path ( in x and 17 in y), it will satisfy the linearized equations... 

A very important example of global bifurcations is the succession of period-doubling transitions, leading from a steady-state solution to time-periodic solutions of increasing period and finally to nonperiodic behavior. This phenomenon has been studied extensively for iterative equations of the form11,12... 

Dymanus [124], However the A-doubling transitions show extensive 14N hyperfine splittings, which we must also take into account. 

We have already discussed the high-resolution spectroscopy of the OH radical at some length. It occupies a special place in the history of the subject, being the first short-lived free radical to be detected and studied in the laboratory by microwave spectroscopy. The details of the experiment by Dousmanis, Sanders and Townes [4] were described in section 10.1. It was also the first interstellar molecule to be detected by radio-astronomy. In chapter 8 we described the molecular beam electric resonance studies of yl-doubling transitions in the lowest rotational levels, and in chapter 9 we gave a comprehensive discussion of the microwave and far-infrared magnetic resonance spectra of OH. Our quantitative analysis of the magnetic resonance spectra made use of the results of pure field-free microwave studies of the rotational transitions, which we now describe. 

The initial studies of Dousmanis, Sanders and Townes [4] were focussed on the hyperfine components of the yl-doubling transitions in the J = 1/2,9/2 and 11/2 rotational levels of the ground 2 n3 /2 state, and the./ = 3/2 and 5/2 levels of the excited 2 n 1/2 state. Figure 10.57 illustrates the yl-doublet and proton hyperfine splitting of the... 

It is also important to remind that double transitivity implies the mapping of all ordered pairs. As an example if the symmetry of the triangle is limited to C3 only, the double transitivity is lost, since this group does not allow odd permutations that are needed to switch the ordering of pairs. As a result the E irrep is split into two complex conjugate one-dimensional irreps. 

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