Collision-free limit

Considerable experimental effort has been aimed at elucidating the collision-free unimolecular dynamics of excited molecules. Processes of interest include the dynamics of highly excited vibrational states, which have been reached by multiphoton absorption, and the various electronic relaxation processes that can occur in electronically excited states of moderate to large molecules, etc. The idealized collision-free limit is approached either by extrapolating data to the limit of zero pressure or by performing experiments in molecular beams. Alternatively, estimates of expected collisional effects are made by using collision cross-sections that are computed from hard-sphere collision rates. These estimates are then utilized to determine whether the experiments are performed in the collision-free domain. 

We begin by assuming that the ion current density to all internal surfaces is equal. Then, if we assume a collision-free sheath with the field being zero at the edge of the positive column, we can express the space charge-limited ion current as16... 

It is clear that, by changing the experimental conditions and/or detection wavelength, limiting values can be found for all of the quantities mentioned above from measurements of the fluorescence decay time. The effects of collisional and spontaneous processes can be separated by conventional Stem—Volmer analysis [36]. The concentration, [M], of quenching molecules is varied and the reciprocal of the observed lifetime is plotted against the concentration of M. The quenching rate coefficient is thus obtained from the slope and the intercept gives the rate coefficient for the spontaneous relaxation processes, which is usually the natural lifetime of the excited state. In cases where the experiment cannot be carried out under collision-free conditions, this is the only way to measure the natural lifetime from observation of the fluorescence decay. 

Thus, in one lifetime (t = 75 pis) the distance which would be traveled is tvp — 2.5 cm. It is thus obvious that serious errors in the measurement of collision-free decay constants could be made if fluorescence cells with dimensions on the order of tdp are used. The limiting dimensions also apply to the observation region since, once the excited molecule moves out of the detection area, the effect is the same as if it had been quenched. Equations for the time dependence of fluorescence which is influenced by migration of long-lived excited molecules to the boundaries of a cylindrical observation region have been developed by Sackett and Yardley. Also included in these equations is the effect of the variation in detection efficiency over the volume of the fluorescence cell. 

Fig. 1 Schematic representation of the pressure effect on the singlet (S(r)=log<(s ip(r)) ) and triplet (/-(r)=log ) content of the excited molecular state. Crosses, collision-free conditions, points and solid line, increasing inert-gas pressure, (a) statistical-limit (b) strongcoupling case (incoherent excitation) (c) strong coupling case (coherent excitation) (d) weak-coupling case (small polyatomics) (e) weak-coupling case (CO,N2).

Changing the acceleration potential or the electrode frequency allows to vary the mass to be detected. From computer simulations ion currents of several lOOnA up to IpA are expected for pure gases and a resolution of m/Am=18 for a separator of 2mm in length and the electrode dimensions as mentioned. Furthermore calculations show that the resolution is limited rather by geometry and available electrode frequency than by thermal motion of the ions. With respect to the mean free path a pressure in the separator below 4Pa will enable a collision free trajectory. 

Definition 3 All the adjacent points in C-space that do not satisfy the Collision-free Constraint are denoted as a real-obstacle, all the adjacent points in C-space that do not satisfy the Closed Chain Constraint are denoted as a closed-chain-obstacle and all the adjacent points in C-space that do not satisfy the Joint Limit Constraint are denoted as a joint-limit-obstacle. A C-space obstacle can be either a real-obstacle a closed-chain-obstacle, or a Joint-limit-obstacle. 

During path planning, the C-point moves along either M-line or the boundary of obstacles. If TC-curve does not intersect with real-obstacles and joint-limit-obstacles, by lemma 3, there is one and only one collision-free segments on TC-curve. If TC-curve intersects real-obstacles and/or joint-limit-obstacles, by lemma 4 and lemma 2, the intersections can be used to represent the collision-free TC-curve segments. 

Corollary 1 If both S and T are in the same 2D-subspace but not in the same free-region, and TC-curve does not intersect the boundaries of joint-limit- and real-obstacles, there is no collision-free path in the C-space. 

Several instniments have been developed for measuring kinetics at temperatures below that of liquid nitrogen [81]. Liquid helium cooled drift tubes and ion traps have been employed, but this apparatus is of limited use since most gases freeze at temperatures below about 80 K. Molecules can be maintained in the gas phase at low temperatures in a free jet expansion. The CRESU apparatus (acronym for the French translation of reaction kinetics at supersonic conditions) uses a Laval nozzle expansion to obtain temperatures of 8-160 K. The merged ion beam and molecular beam apparatus are described above. These teclmiques have provided important infonnation on reactions pertinent to interstellar-cloud chemistry as well as the temperature dependence of reactions in a regime not otherwise accessible. In particular, infonnation on ion-molecule collision rates as a ftmction of temperature has proven valuable m refining theoretical calculations. 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

This is an indication of the collective nature of the effect. Although collisions between hard spheres are instantaneous the model itself is not binary. Very careful analysis of the free-path distribution has been undertaken in an excellent old work [74], It showed quite definite although small deviations from Poissonian statistics not only in solids, but also in a liquid hard-sphere system. The mean free-path X is used as a scaling length to make a dimensionless free-path distribution, Xp, as a function of a free-path length r/X. In the zero-density limit this is an ideal exponential function (Ap)o- In a one-dimensional system this is an exact result, i.e., Xp/(Xp)0 = 1 at any density. In two dimensions the dense-fluid scaled free-path distributions agree quite well with each other, but not so well with the zero-density scaled distribution, which is represented by a horizontal line (Fig. 1.21(a)). The maximum deviation is about... 

The orientation of linear rotators in space is defined by a single vector directed along a molecular axis. The orientation of this vector and the angular momentum may be specified within the limits set by the uncertainty relation. In a rarefied gas angular momentum is well conserved at least during the free path. In a dense liquid it is a molecule s orientation that is kept fixed to a first approximation. Since collisions in dense gas and liquid change the direction and rate of rotation too often, the rotation turns into a process of small random walks of the molecular axis. Consequently, reorientation of molecules in a liquid may be considered as diffusion of the symmetry axis in angular space, as was first done by Debye [1],... 

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... 

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