Boltzmann rotational populations

If the spectrum is observed in emission it is the rotational populations in the upper state which determine relative intensities. They may or may not be equilibrium Boltzmann populations, depending on the conditions under which the molecule got into the upper state. 

The rotational population distributions were Boltzmann in nature, characterized by 7Ji = 640 35 K. This seems substantially lower than yet somewhat larger than the temperature associated with the translational degree of freedom. The lambda doublet species were statistically populated. The population ratio of i =l/t =0 was roughly 0.09, consistent with a vibrational temperature Ty— 1120 35K. The same rotational and spin-orbit distributions were obtained for molecules desorbed in t = 1 as for f = 0 levels. Finally, there was no dependence in the J-state distributions on desorption angle. 

Figure 3.15. Rotational state distributions of NO produced in direct scattering from Ag(lll) at Ts 600 as a function of incident normal energy En. Rotational populations Nj are plotted in such a way that a Boltzmann distribution characterized by a temperature T is a straight line. The different symbols correspond to rotational populations derived from the different rotational transitions as listed. From Ref. [160].

Using a tunable laser as a probe they have observed that CN(X2E) radicals produced at this wavelength are vibrationally and rotationally excited. The rotational distribution follows the Boltzmann law, indicating that dissociation is not immediate but occurs after many vibrations of the electronically excited molecule. Thus, the distribution of the rotational population reflects (lie statistical nature of the dissociation processes. The distribution of the excess energy beyond that required to break the C—C bond is 54% in electronic, 20% in translational, 14% in vibrational, and 11% in rotational energies. See also p. 87. 

A CARS experiment has recently been done to determine the amount of vibrational and rotational excitation that occurs in the O2 (a- -A) molecule when O3 is photodissociated (81,82). Valentini used two lasers, one at a fixed frequency (266 nm) and the other that is tunable at lower frequencies. The 266 nm laser light is used to dissociate O3, and the CARS spectrum of ( (a A), the photolysis product, is generated using both the fixed frequency and tunable lasers. The spectral resolution (0.8 cm l) is sufficient to resolve the rotational structure. Vibrational levels up to v" = 3 are seen. The even J states are more populated than the odd J states by some as yet unknown symmetry restrictions. Using a fixed frequency laser at 532 nm (83) to photolyze O3 and to obtain the products 0(3p) + 02(x3l g), a non-Boltzmann vibrational population up to v" = k (peaked at v" = 0) is observed from the CARS spectrum. The rotational population is also non-Boltzmann peaked at J=33, 35 33, 31 and 25 for v" = 0,1,2,3, and k, respectively. Most of the available energy, 65-67%, appears in translation 15-18% is in rotation and 17-18% is in vibration. A population inversion between v" = 2 and 3 is also observed. 

Rotational Population Distributions. As expected, there is an overpopulation in the A state level and an underpopulation in the X state level connected by the laser, compared to a thermal distribution (see Fig. 1). Higher rotational levels (N 6) in the A-state are described by a Boltzmann distribution with T 940°K, well less than the gas temperature and reflecting the energy dependence of the or. The high-N levels of the X-state are described by a Boltzmann distribution with very high T (3200°K) but this may be an artifact of the model, due principally to the assumed... 

They found that the rotational populations of high and low rotational levels of the e n state were expressed by double Boltzmann temperatures of 903 and 492 K at 1.4 Torr and 878 and 278 K at 250 Torr, respectively. Collins and Robertson examined the axial variation and pressure dependence of the intensity of the He2 emission and HeJ concentration, and identified the dominant populating meehanism of He2 to be recombination process 27. 

The second differences may be measured directly from the spectrum without any knowledge whether J increases or decreases as one follows a branch toward higher frequency. Thus one immediately obtains a good estimate for B — B" neglecting contributions from D and D". The line intensities usually vary smoothly along a rotational branch because collisions cause rotational populations to approach Boltzmann equilibrium rapidly and rotational linestrength factors (see Section 6.1 and Whiting, et ai, 1980) are approximately linear in J. 

The energy separations between rotational levels in a diatomic molecule in a given vibrational and electronic state are typically small compared with the thermal translational energy. Nearly all gas kinetic collisions produce a change in the rotational quantum number, whereas collisions producing a change in the vibrational and electronic quantum numbers occur much less frequently. Consequently, the relative rotational population distribution in a sufficiently long-lived vibrational state has a Boltzmann distribution with a rotational temperature that reflects the gas kinetic temperature. The relative emission or absorption intensity is then... 

To allow us to extract rotational populations from vibrational bands like the v = 5 band of Figure 3, we have written a caiputer code which solves these n coupled non-linear equations. Figure 6 illustrates the results obtainable with this code, ihe calculated and observed CARS spectra are nearly identical. The rotational populations derived from the experimental spectrum by the deconvolution program give a linear Boltzmann plot with a temperature of 299 K and a correlation coefficient of 0.997. Ihe experimental spectrum was taken at an ambient temperature of 295 K. 

To make quantitative statements about the product internal distribution a computer program is utilized to simulate the observed excitation spectrum [10]. As input for the calculations we estimate the relative vibrational and rotational populations. Each line is weighted by the population of the initial (v, J ) level, by the Franck-Condon factor and the rotational line strength of the pump transition. At each frequency, the program convolutes the lines with the laser bandwidth and power to produce a simulated spectrum such spectra are compared visually with the observed spectra and new estimates are made for the (v ,J") populations. Iteration of this process leads to the "best fit" as shown in the lower part of Fig. 3. For this calculated spectrum all vibrational states v" = 0...35 are equally populated as is shown in the insertion. The rotation, on the other hand, is described by a Boltzmann distribution with a "temperature" of 1200 K. With such low rotational energy no band heads are formed for v" < 5 in the Av = 0 sequence and for nearly all v" in the Av = +1 sequence (near 5550 A). 

Some work also has been done using the "measured relaxation" technique (18,33,40,41) which can give initial vibrational distributions and relative total rate constants for different RH molecules. This technique usually Involves a flowing afterglow apparatus with bulk flow velocity in the 100 m sec regime (23,, ). In these experiments the reactants are mixed at various distances, which can be converted to times, upstream of an observation window. The observed rotational populations are relaxed to a Boltzmann distribution however, the vibrational relaxation can be followed as a function of time and can be extrapolated to obtain the initial populations at zero time. 

The degeneracy factor (27 + 1) in Eq. (3.5.11) has a strong influence on the rotational population distribution within a vibrational level. At a particular temperature thedegeneracy factorcausesthepopulationtobepioportionaltoftcB/fcr at7 =0, and to increase for small 7. As 7 increases further the effect of the exponential form of the Boltzmann factor becomes more and more dominant, causing the population to approach zero for high 7 values. Figure 3.5.1 shows this distribution of rotational... 

NH in its v = 1 vibrational level and in a high rotational level (e.g. J> 30) prepared by laser excitation of vibrationally cold NH in v = 0 having high J (due to nahiral Boltzmann populations), see figure B3.T3 and... 

There is a stack of rotational levels, with term values such as those given by Equation (5.19), associated with not only the zero-point vibrational level but also all the other vibrational levels shown, for example, in Figure 1.13. However, the Boltzmann equation (Equation 2.11), together with the vibrational energy level expression (Equation 1.69), gives the ratio of the population of the wth vibrational level to Nq, that of the zero-point level, as... 

The intensity distribution among rotational transitions in a vibration-rotation band is governed principally by the Boltzmann distribution of population among the initial states, giving... 

The intensity distribution among the rotational transitions is governed by the population distribution among the rotational levels of the initial electronic or vibronic state of the transition. For absorption, the relative populations at a temperature T are given by the Boltzmann distribution law (Equation 5.15) and intensities show a characteristic rise and fall, along each branch, as J increases. 

Depending on the method of pumping, the population of may be achieved by — Sq or S2 — Sq absorption processes, labelled 1 and 2 in Figure 9.18, or both. Following either process collisional relaxation to the lower vibrational levels of is rapid by process 3 or 4 for example the vibrational-rotational relaxation of process 3 takes of the order of 10 ps. Following relaxation the distribution among the levels of is that corresponding to thermal equilibrium, that is, there is a Boltzmann population (Equation 2.11). 

The rotational temperature is defined as the temperature that describes the Boltzmann population distribution among rotational levels. For example, for a diatomic molecule, this is the temperature in Equation (5.15). Since collisions are not so efficient in producing rotational cooling as for translational cooling, rotational temperatures are rather higher, typically about 10 K. 

Figure 10.7 The population of excited energy states relative to that of the ground state for the CO molecule as predicted by the Boltzmann distribution equation. Graph (a) gives the ratio for the vibrational levels while graph (b) gives the ratio for the rotational levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations. The dots represent ratios at integral values of v and 7. which are the only allowed values.

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