Thermal distribution

Figure 4.29. Sample assembly for optical shock temperature measurements. The sample consists of a metal film deposited on a transparent substrate which serves as both an anvil and a transparent window through which thermal radiation is emitted. Rapid compression of gases and surface irregularities at the interface between the sample film and the driver produce very high temperatures in this region. The bottom portion of the figure illustrates the thermal distribution across through the assembly. (After Bass et al. (1987).)...

Thus, for a transition between any two vibrational levels of the proton, the fluctuation of the molecular surrounding provides the activation. For each such transition, the motion along the proton coordinate is of quantum (sub-barrier) character. Possible intramolecular activation of the H—O chemical bond is taken into account in the theory by means of the summation of the probabilities of transitions between all the excited vibrational states of the proton with a weighting function corresponding to the thermal distribution.3,36 Incorporation in the theory of the contribution of the excited states enabled us in particular to improve the agreement between the theory and experiment with respect to the independence of the symmetry factor of the potential in a wide region of 8[Pg.135]

V 1s n) is the normalized thermal distribution of configurations of the distinguished molecule in isolation [10], i.e., the required marginal distribution. The remaining set of brackets here indicates the average over solvent coordinates. The second set of brackets are not written on the right here because the averaging over solute coordinates is explicitly written out. This last formula is... 

FIGURE 8.5 Variation of the b-value of the gaussian thermalization distribution in n-hexane with excess electron energy. While the b values were obtained by comparison with experimental results, other forms of initial distribution are also possible. Reproduced from Hirata and Mataga (1991). 

Taking Ay = 15 nm, . = 5 eV, and other values as before, the b value for LAr is evaluated as 1400 nm, which is much larger than 133 nm, obtained by fitting the free-ion yield to the Onsager formula (vide supra). Similar calculations for LKr and LXe give b values of the gaussian thermalization distribution... 

The width of the peaks in LETS depends upon the sharpness of the onset of the inelastic process, which in turn depends upon the thermal distribution of electron energies about EP. Thus, the IETS line width depends strongly on temperature and as shown by (3) [75]. Because of this, vibrational IETS provides infrared-quality resolution only when performed below 5 K. Electronic transitions are usually much broader than vibrational transitions therefore, electronic IETS is usually performed at liquid nitrogen temperature and slightly above (>77 K). An example of a system showing both vibrational and electronic IETS is presented in Fig. 5 [19]. 

Notice that the thermal average can be given by taking the vacuum average 0,0) of a thermal non-tilde variables. For instance for the particular case of the bosonic number operator, n = a a, the thermal distribution, as in Eq.(7) reads,... 

The desorption flux is so low under these conditions that no gas phase collisions occurred between molecular desorption and LIF probing. Phase space treatments " of final-state distributions for dissociation processes where exit channel barriers do not complicate the ensuing dynamics often result in nominally thermal distributions. In the phase space treatment a loose transition state is assumed (e.g. one resembling the products) and the conserved quantities are total energy and angular momentum the probability of forming a particular flnal state of ( , J) is obtained by analyzing the number of ways to statistically distribute the available (E, J). 

Thermal distributions of NO(u 2, J, A Ej,) states were observed, wherein the population in any level was determined by the internal energy and the parameter Tr. and independent of spin-orbit state or lambda doublet species. This is in contrast to the rotational rainbows, the propensities for preferential population in the Il(A ) lambda doublet species and the Fj spin orbit state which were observed in direct inelastic scattering of NO/Ag(l 11). 

The relative population ratio FJFi was slightly higher than expected from a 300 K thermal distribution (e.g. 2.1 vs 1.8). Of particular note, in comparison to a simple Boltzmann distribution, there was a substantial absence of population in the F2(J < S.S) levels from that expected based on a thermal (300 K) distribution. Approximately 1% of the desorbed molecules were vibrationally excited. 

The dynamics of the normal mode Hamiltonian is trivial, each stable mode evolves separately as a harmonic oscillator while the imstable mode evolves as a parabolic barrier. To find the time dependence of any function in the system phase space (q,pq) all one needs to do is rewrite the system phase space variables in terms of the normal modes and then average over the relevant thermal distribution. The continuum limit is introduced through use of the spectral density of the normal modes. The relationship between this microscopic view of the evolution... 

In a realistic simulation, one initiates trajectories from the reactant well, which are thermally distributed and follows the evolution in time of the population. If the phenomenological master equations are correct, then one may readily extract the rate constants from this time evolution. This procedure has been implemented successfully for example, in Refs. 93,94. Alternatively, one can compute the mean first passage time for all trajectories initiated at reactants and thus obtain the rate, cf. Ref 95. 

For E < 0, let f(E)dEdt denote the probability to find the system within the time interval dt, with a mode energy between E and E + dE at the barrier of the p mode. For a thermal distribution W, near the barrier top feq(E) =... 

Figure 10.8 Examples of controlling rotational wave packets with an optical centrifuge. Shifting the center of a narrow wave packet is shown for centrifuged oxygen on (a), whereas varying the width of the wave packet is demonstrated for nitrogen on (b). Black dots represent the thermal distribution at room temperature.

For molecular desorption, laser spectroscopic studies of the desorbing molecule can give full internal state distributions, Df Ef, 6f, v, J, f M ), Ts), where f M ) is some distribution function describing the rotational orientation/alignment relative to the surface normal. For thermal desorption in non-activated systems, most atoms/molecules have only modest (but important) deviations from a thermal distribution at Ts. However, in associative desorption of systems with a barrier, the internal state distributions reveal intimate details of the dynamics. Associative desorption results from the slow thermal creation of a transition state, with a final thermal fluctuation causing desorption. Partitioning of the energy stored in V into... 

Figure 3.32. H2 Sticking (dissociative adsorption) probability S on Pd(100) as a function of incident normal kinetic energy Et = En. Circles are experiment [304], dashed and solid line are 6D first principles quantum dynamics with H2 in the ground state and a thermal distribution appropriate to the experiments, respectively [109]. The inset is also 6D first principles quantum dynamics but based on a better PES [309]. From Ref. [2].

The theoretical model developed to explain these experiments is based on inelastic tunneling of electrons from the tip into the 2ir adsorbate resonance that induces vibrational excitation in a manner similar to that of the DIMET model (Figure 3.44(b)). Of course, in this case, the chemistry is induced by specific and variable energy hot electrons rather than a thermal distribution at Te. Another significant difference is that STM induced currents are low so that vibrational excitation rates are smaller than vibrational de-excitation rates via e-h pair damping. Therefore, coherent vibrational ladder climbing dominates over incoherent ladder climbing,... 

For a thermal distribution at 298 K, determine the ratio of populations of molecules in the 7 = 3 and j = 2 rotational energy levels. 

Does the presence of a thermal distribution of initial states of the reactant molecule seriously compromise the extent of possible control of the molecular dynamics ... 

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