Transform limited collisions

While collisional resonances 5 MHz wide are interesting for spectroscopic purposes, what makes them most interesting is that the 5 MHz line width implies that the collision lasts at least 200 ns, a time not much less than the 1 fxs period allowed for the collisions to occur. If the collision linewidths can be reduced to the inverse of the time allowed for the collisions to occur, the collisional resonances become transform limited, and we know when each collision begins and ends. 

Thomson et al. reached the transform limit by incorporating two improvements.21 First, Helmholtz coils were placed around the interaction region to cancel the earth s magnetic field. Second, the time interval allowed for the collisions had previously been defined by the laser pulse and the field ionization 

Ramsey, Molecular Beams (Oxford University Press, London, 1956). 

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The two extreme cases cu 1/r and cu -C 1/t are easily understood, but in very different terms, so it is less clear how to think about the case w 1/r. Nonetheless, the low frequency regime suggests that the phase of the rf field relative to when the collision occurs is likely to be important. To control this phase we use transform limited collisions of /C atoms in a velocity selected beam so that we know when the collisions begin and end, and we synchronize the rf field to the laser pulse initiating the collisions [Renn 1991], Specifically, we have studied the process of Eq. (10) in rf fields phase locked to the collisions. 

We now examine whether an enantiomeric excess S, i.e., the difference between the number of D and L molecules, can result from the irradiation of a 50 50 racemic mixture by any combination of transform limited light pulses in the absehee of collisions. We have that... 

Elastic collisions are chemically inert no chemical bonds are broken or formed. Our interest in elastic collisions will usually be limited to problems involving molecular transport (the movement of molecules from one place to another, without any chemical transformation). Elastic collisions are also useful for illustrating the following important point about chemical bond formation. 

The impact theory defines uniquely the spectral transformation in the limit of weak collisions. Expanding in a series over J — J the integrand of Eq. (6.4) one can obtain at T = /coq%j 1... 

Without essential limitation of generality it may be assumed that the orientation of the molecule and its angular momentum are changed by collision independently, therefore F(JU Ji+, gt) = f (Jt, Ji+i)ip(gi). At the same time the functions /(/ , Ji+ ) and xp(gi) have common variables. There are two reasons for this. First, it may be due to the fact that the angle between / and u must be conserved for linear rotators for any transformation. Second, a transformation T includes rotation of the reference system by an angle sufficient to combine axis z with vector /. After substitution of (A7.16) and (A7.14) into (A7.13), one has to integrate over those variables from the set g , which are not common with the arguments of the function / (/ , /j+i). As a result, in the MF operator T becomes the same for all i and depends on the moments of tp as parameters. 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

This equation is readily transformed to an integral equation for different from i and in <— k,- Y(z] — k )) never appear in two successive collision operators because otherwise we would get a negligible contribution in the limit of an infinite system moreover as these dummy particles have zero wave vectors in the initial state, they have a Maxwellian distribution of velocities (see Eq. (418)). This allows us to write Eq. (A.74) in the compact form ... 

The main lines of the Prigogine theory14-16-17 are presented in this section. A perturbation calculation is employed to study the IV-body problem. We are interested in the asymptotic solution of the Liouville equation in the limit of a large system. The resolvent method is used (the resolvent is the Laplace transform of the evolution operator of the N particles). We recall the equation of evolution for the distribution function of the velocities. It contains, first, a part which describes the destruction of the initial correlations this process is achieved after a finite time if the correlations have a finite range. The other part is a collision term which expresses the variation of the distribution function at time t in terms of the value of this function at time t, where t > t t—Tc. This expresses the fact that the system has a memory because of the finite duration of the collisions which renders the equations non-instantaneous. 

Estimation of rates for redox reactions in environmental systems requires that the problem be formulated in terms of specific oxidation and reduction half-reactions. In addition, we assume that the rate-limiting step of the transformation mechanism is bimolecular—that is, the slow step requires an encounter (collision) between the electron donor and electron acceptor. Under most conditions found in environmental systems, such reactions exhibit rate laws for the disappearance of a pollutant, P, that are first-order in concentration of P and first-order in the concentration of environmental oxidant or reductant, E,... 

Comparing FTMS with Fourier transform nuclear magnetic resonance (FTNMR), we first notice how the frequency range to be covered here is very large. Second, relaxation in NMR is invariably linked with the interaction among liquid-phase or solid-phase molecules. In the gas phase, relaxation depends on the vacuum and on the stability of the ions being observed. If the vacuum is not sufficient, collisions slow the ions and their movement becomes incoherent. The observation of an ion is also limited to its lifetime. 

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