Non-Boltzmann distributions

Clearly, if a situation were achieved such that exceeded Np, the excess energy could be absorbed by the rf field and this would appear as an emission signal in the n.m.r. spectrum. On the other hand, if Np could be made to exceed by more than the Boltzmann factor, then enhanced absorption would be observed. N.m.r. spectra showing such effects are referred to as polarized spectra because they arise from polarization of nuclear spins. The effects are transient because, once the perturbing influence which gives rise to the non-Boltzmann distribution (and which can be either physical or chemical) ceases, the thermal equilibrium distribution of nuclear spin states is re-established within a few seconds. 

The above derivation shows that Jarzynski s identity is an immediate consequence of the Feynman-Kac theorem. This connection has not only theoretical value, but is also useful in practice. First, it forms the basis for an equilibrium thermodynamic analysis of nonequilibrium pulling experiments [3, 15]. Second, it helps in deriving a Jarzynski identity for dynamics using thermostats. Moreover, this derivation clarifies an important aspect trajectories can be thought of as mapping initial conditions (I = 0) to trajectory endpoints, and the Boltzmann factor of the accumulated work reweights that map to give the desired Boltzmann distribution. Finally, it can be easily extended to transformations between steady states [17] in which non-Boltzmann distributions are stationary. 

Here, w(xfc) is the weighting factor for any property at a given position on the fcth step xfc. For example, for a constant-temperature molecular dynamics or a Metropolis MC run, the weighting factor is unity. However, we wish to leave some flexibility in case we want to use non-Boltzmann distributions then, the weighting factor will be given by a more complicated function of the coordinates. The ergodic measure is then defined as a sum over N particles... 

Just as above, we can derive expressions for any fluorescence lifetime for any number of pathways. In this chapter we limit our discussion to cases where the excited molecules have relaxed to their lowest excited-state vibrational level by internal conversion (ic) before pursuing any other de-excitation pathway (see the Perrin-Jablonski diagram in Fig. 1.4). This means we do not consider coherent effects whereby the molecule decays, or transfers energy, from a higher excited state, or from a non-Boltzmann distribution of vibrational levels, before coming to steady-state equilibrium in its ground electronic state (see Section 1.2.2). Internal conversion only takes a few picoseconds, or less [82-84, 106]. In the case of incoherent decay, the method of excitation does not play a role in the decay by any of the pathways from the excited state the excitation scheme is only peculiar to the method we choose to measure the fluorescence (Sections 1.7-1.11). 

NO (v = 0, 2 = 1/2 and 3/2) are measured with a probe laser beam-sample distance of 1.65 mm at a delay time of 3.5 pis (velocity of 0.47 km/s), and shown in Fig. 14, which reveals a non-Boltzmann distribution. Furthermore, the two spin-orbit states exhibit inversion population, the 2 = 3/2 state being more populated. Since the RAIRS spectmm shows that the peak intensity at 1700 cm-1 decreases with laser irradiation, it is concluded that desorption of the on-top species occurs. 

Table 7. Observed non-Boltzmann distributions within the rotational levels of several interstellar molecules...

The formation of vibrationally excited products is nearly always energetically possible in an exothermic reaction, and these products can be detected by observing either an electronic banded system in absorption or the vibration-rotation bands in emission. In principle, rotational level distributions may be determined by resolving the fine structure of these spectra, but rotational energy is redistributed at almost every collision, so that any non-Boltzmann distribution is rapidly destroyed and difficult to observe. In contrast, simple, vibrationally excited species are much more stable to gas-phase deactivation and the effects of relaxation are less difficult to eliminate or allow for. 

Where an atom has a multiplet ground state, reaction may populate these sublevels with a non-Boltzmann distribution. This is difficult to observe since for light atoms the spin-orbit splitting is small and relaxation is rapid, and also because optical transitions between the components of the multiplet are strongly forbidden. Absorption measurements are possible but have scarcely been applied at all to this particular problem. 

Figure 1.12 Rotational level distributions in HC1 (v = 2) from the H + Cl2 and Cl + HI reactions [264], The abscissa records both J, the rotational quantum number, and Fj, the fraction of available energy present as rotation. The arrows indicate the limits of excitation determined by the fact that E (v = 2) + Zi(./)cannot exceed Q, the total available energy. The broken lines indicate best fit Boltzmann distributions and show that the majority of the rotators are in a highly non-Boltzmann distribution. The subsidiary peaks at low J conform to a low-temperature Boltzmann distribution.

Discussion of the these three methods is outside the scope of this book, but in later chapters we consider other methods for producing much less dramatic non-Boltzmann distributions. By using rf irradiation to alter spin populations, the nuclear Overhauser effect results in signal enhancement (Chapters 8 and 10). Several techniques use pulse sequences to transfer polarization from nuclei with large y to nuclei with small y in solids (Chapter 7) and liquids (Chapters 9 and 12), hence to provide significant signal enhancement. 

Gaussian-like distribution of energy around the energy average. Other ensembles with non-Boltzmann distributions can enhance the sampling considerably for example, in the multi-canonical approach [97, 98], all the conformations are equiprobable in energy in Tsallis statistics [99], the distribution function includes Boltzmann, Lorentzian, and Levy distributions. 

In addition to the S( S) signal, Donovan (109) detected absorption due to vlbrationally excited (v" = 2) CO and S(3p2, Pj and Pq)> the latter in a non-Boltzmann distribution which does not appear as quickly following the flash lamp discharge as does the S( S) absorption. This delayed appearance of the S( Pj) atoms suggests that they are, at least in part, produced in reactions subsequent to the photolysis. This is confirmed by Donovan et al. (Ill) who studied the flash photolysis with added He as the buffer gas rather than with added Ar. The strong spectrum of S( Pj) observed in the presence of Ar at short delay was virtually absent in the presence of He. Thus it seems that process 16a is not an important process at the wavelengths employed by Donovan. 

In Fig. 22a the arrows mark the onset of a Boltzmann tail for 1% and 10% of atomic nitrogen. The corresponding t> s have been calculated according to Ref.8). The occurrence of non-Boltzmann distributions of the type illustrated in Figs. 19 and 22 has been demonstrated experimentally. 

Direct determination of non-Boltzmann distributions of the vibrational levels of the ground N2(X1S ) state has recently been performed by a new diagnostic technique, coherent anti-Stokes Raman spectroscopy21 with results consistent with calculated distributions. Kinetic data on N2 dissociation amenable to a comparison with theoretical predictions are scanty. Data from Ref.223 can however be quoted and are reported in Fig. 26. The dashed line has been calculated on the assumption that dissociation takes place via predissociated electronic states excited by direct electron impact. Observed dissociation rates are higher and a much better agreement has been claimed with dissociation rates calculated on the basis of a pure vibrational mechanism223 22b. ... 

The kinetics of dissociation of polyatomic molecules, in particular CH4, stimulated by vibrational excitation in conditions of not very high non-equilibrium parameters y = (Tv — To)/To has been analyzed by Kuznetsov (1971). The CH4 dissociation (9-13) proceeds through vibrational excitation of CH4 molecules at any parameters y = (Tv - To)/Tq. The vibrational energy distribution is an essentially non-Boltzmann distribution in this case, and it is characterized by both temperatures, Tv and To, even when Tv > Tq. The rate coefficient kR(To, Tv) of the methane dissociation (9-13) in non-equihbrium conditions (Tv > To) can be expressed as follows (Kuznetsov, 1971) ... 

Optical electron spin polarisation (OEP) is the term used to describe a non-Boltzmann distribution of the populations of the three zero-field or Zeeman components of an optically-excited triplet state. This non-thermal equilibrium can be a stationary or a non-stationary state. The optical excitation, that is e.g. the UV excitation, must be neither narrow-band nor polarised, and at low temperatures, OEP is the normal case for most triplet states in organic tt-electron systems. The OEP is... 

The extremely narrowband emission of a laser allows the specific excitation of molecular states. The non-Boltzmann distribution produced by the excitation process is quickly destroyed by radiation processes and collisional deactivation. The relative contribution of these different deactivation channels depends on the nature of the level excited as shown in Fig. 3. In the microwave region where rotational levels are excited, the radiative life time is very long compared to the very efficient rotational relaxation processes (R—R rotation—rotation transfer and R—T rotation—translation transfer). Therefore, the absorbed radiation energy is transformed within a few gas kinetic collisions into translational energy. The situation is similar for... 

Fig. 6-3 Energy level schemes for a four-level double resonance experiment. The molecules are pumped from level 2 to level 1 by intense radiation with frequency Vp. The non-Boltzmann distribution is then transferred to other levels by collision, and the transfers are monitored by a weak intensity signal with frequency Vs- MW designates microwave and IR infrared radiation. (Adapted from Oka [11].)...

There remains some non-Boltzmann distribution in the higher rotational levels but, as shown in Fig. 7, if the pressure in the reservoir is... 

Our analysis indicates high rotational non-Boltzmann distribution peaking in high rotational levels (J" 31. The different vibrational levies of the NO fragment show similar rotational distributions. Identical distributions were derived for each dissociating wavelength showing that the rotational state distribution is unaffected by the different initial... 

The second point that should be made is that the situation of isolated particles in the high vacuum—in particular the ill-definition of temperature and the non-Boltzmann distribution of internal energies—may appear to be somewhat uncommon for many chemists who are used to do chemistry in solution. Nevertheless, if one considers the great new insights into, for example, the intramolecular reactivity of supermolecules, it is certainly worthwhile even for solution-phase supramolecular chemists to invest effort and time into mass spectrometric analyses of their supermolecules. 

A last increasingly popular family of MC simulations, not to be confused with the Metropolis method, exploits sampling from non-Boltzmann distributions to simulate kinetic events which are extremely rare compared to the typical molecular timescales, e.g., as is the case of charge transfer (dynamic or Kinetic MC [105], KMC) or reactive events (Gillespie s stochastic simulation algorithm [106]). 

Method 2 uses an alternative approach of continuous intramolecular potentials instead of RIS probabilities and and this has been applied to the PE I model of polyethylene chains with N = 1000. It has been pointed out, however, that site-by-site chain growth with excluded volume samples from a non-Boltzmann distribution of end-to-end distances. In addition the effective density increases during growth so this procedure also gives a... 

The population distribution between the two spin states is conveniently described by the concept of a spin temperature. For a Boltzmann distribution Tg T however, for a non-Boltzmann distribution Tg Ti but approaches Tl in the spin-lattice relaxation time. See Ref. 21 for details. 

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