The Landau-Teller theory

The metastability of vibrational energy results from two factors (a) the amplitude of vibration is usually small compared to the range of the coordinate x for which the molecules experience each others repulsive forces, and (b) the period of vibration is usually short compared to the duration of the interaction during collision. Landau and Teller3 pointed out that the probability of energy transfer depends on and increases with the ratio of the period of vibration to the duration of the collision. If the repulsive part of the interaction potential is shallow, the forces during the collision tend to act on the centres of gravity, rather than on 

The quantity / is usually called the characteristic length of the interaction potential, and is also employed in the more realistic wave-mechanical treatment. Many authors employ the symbol, a, which is equal to /-1. The form of the molecular interaction potential can be determined in terms of a suitable model, from experimental measurements of the temperature dependence of the viscosity of a gas, whence the characteristic length can be estimated. 

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This asymptotic form is plotted in Fig. 5. A feature of BBM(d>) is that it decreases asymptotically with frequency to zero. If the atom B is involved in vibrational motion at frequency oo (Oq, the coupling with the bath through binary collisions is small and the slow dissipation is the stochastic manifestation of slow vibrational relaxation. The most significant feature of Eq. (3.17) is that the dependence in the exponent of Eq. (3.17) is equivalent to an exponent This is just the form of the Landau-Teller theory of vibration-translation (V-T) energy transfer in atom-diatom collisions, and this form is almost universally used to fit vibrational relaxation rates in such systems. This will be dealt with in more detail in Section V C. The utility of BBM(d>) is that it pertains to atom-atom collisions in which the atom B is bonded to the other atoms by arbitrary potentials. No assumptions have been made about the intramolecular motions, although the use of BBM(d)) implies linear coupling to the displacements of atom B. Grote et al. have alluded to the form of Eq. (3.20) for di = 0 in a footnote. 

For a comparison of the assumptions underlying our theory of liquid phase VER and those of the Landau-Teller theory see sec. 23 of ref. 19. 

The most simple way to accomplish this objective is to correct the external field operator post factum, as was repeatedly done in magnetic resonance theory, e.g. in [39]. Unfortunately this method is inapplicable to systems with an unrestricted energy spectrum. Neither can one use the method utilizing the Landau-Teller formula for an equidistant energy spectrum of the harmonic oscillator. In this simplest case one need correct... 

E.E.Nikitin, The Landau-Teller model in the theory of collisional excitation of molecules, Proceedings of second USSR summer school on physics of electronic and atomic collisions. Leningrad Physico-Technical Institute, 1973, p.l4... 

The Landau-Teller formula was generalized in SSH theory (Schwartz, Slawsky, Herzfeld, 1952) and, in particular, by Bilhng (1986). A semi-empirical numerical relation... 

Thus, for many molecules the simple SSH theory based on the Landau-Teller model [261] appears to provide qualitatively the mass (p), frequency (co) and temperature (T) dependences of (Pi,o) explain the vast variation of (Pi,o)... 

G. Deviations from the Landau-Teller temperature dependence, Figure 9.10. Estimate the lifetime of, say, an HCl(v = 0) HCl(v =1) van der Waals dimer, using RRK theory, and argue that at low temperatures it will be long enough for a sufficient number of internal collisions to take place and relax HCl(v = 1). 

This is the famous result of Landau and Teller. In many of the more detailed theories, to be discussed below, this result appears explicitly as the principal contribution to the dependence of transition probability on temperature. In practical units, equation (6) is frequently written in the form ... 

We can thus summarize the necessary conditions for an efficient energy transfer (i.e., a resonance function near unity). At a given energy gap, the resonance function 7 ( ) will be exponentially small at low velocities, and will increase with increasing velocities. This version is the form most useful for colhsion theory. An indirect illustration of this exponential gap principle is the strong positive temperature dependence of the V—T relaxation rate. For most diatomic molecules the temperature dependence is best fitted 1 a so-called Landau-Teller equation... 

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