Kramers theory

In Equation 24, [F], [D], [R] denote the field, diffusion, and recombination terms, respectively. If we take the initial distribution of ion densities given by Equation 25, then the relative importance of the three terms can be evaluated for r = b as 

For a-particles the linear ionization density is approximately N = 4 x lO cm- In addition, at room temperature, we have = 2, kgT = 0.03 eV and b = 10 cm. With these data we obtain (F)/(R) = 0.6 x 10 E (E in V/cm) and (D)/(R) = 0.014. In the track, recombination is the main process. These conditions induced Kramers (1952) to solve Equation 24 by at first neglecting the diffusion term. Then the diffusion term was included. At high fields, he found the ionization yield, G, to depend on E as 

Gjot is the total yield of ionization. It is usually assumed to be the same as in the gas phase. An extrapolation of the ionization current of the liquid is made by plotting ijon vs. 1/E. Thomas and Imel (1987) extended Kramers reasoning to the case of liquid argon or xenon. Here the diffusion of the positive ions can be neglected when compared to the diffusion coefficient of the electrons (see Chapter 3). Setting = 0 simplifies Equation 24 to 

With the assumption that each electron/ion pair can be considered as an isolated pair, these equations were solved analytically, giving for the fraction of charge collected. 

Here a denotes the dimension of a box which contains n+(t = 0) carrier pairs. 0 

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The transition from k to on the low-pressure side ean be eonstnieted using iiiidtidimensional unimoleeular rate theory [1, 44], if one knows the barrier height for the reaetion and the vibrational frequeneies of the reaetant and transition state. The transition from to k y ean be deseribed in temis of Kramers theory [45]... 

Larson R S and Kostin M D 1982 Kramers theory of chemical kinetics curvilinear reaction coordinates J. Chem. Phys. 77 5017-25... 

Berezhkovskii A M and Zitserman V Yu 1993 Multi-dimensional Kramers theory of the reaction rate with highly... 

Nitzan A 1988 Activated rate processes in condensed phases the Kramers theory revisited Adv. Chem. Phys. 70 489 Onuchic J N and Wolynes P G 1988 Classical and quantum pictures of reaction dynamics in condensed matter resonances, dephasing and all that J. Phys. Chem. 92 6495... 

From these potential energy curves, the reaction rate can be calculated with the aid of Kramers theory. In the limit of a high solvent friction y, the rate is given by Kramers [1940] and Zusman [1980]... 

If friction plays a role in the crossing of the energy barrier, the reaction is slower than predicted by transition-state theory. According to Kramers theory [20] the preexponential factor must then be replaced by ... 

To appreciate this latter point, we consider four important limits for the GH theory [1,21,221. First, if the adjustment of the solvent is rapid on the time scale of A 1, then the frequency dependence of (A) can be safely ignored, and the GH equations reduce to the famous Kramers Theory result [20]... 

In this nonadiabatic limit, the transmission coefficient is determined, via (2.8) by the ratio of the nonadiabatic and equilibrium barrier frequencies, and is in full agreement with the MD results [5a-5c]. (By contrast, the Kramers theory prediction based on the zero frequency friction constant is far too low. Recall that we emphasized for example the importance of the tail to the full time area of the SN2 (t). In the language of (3.14), the solvation time xs is not directly relevant in determining... 

S. Jun, J. Bechhoefer, andB.-Y. Ha, Diffusion-limited loop formation of semiflexible polymers Kramers theory and the intertwined time scales of chain relaxation and closing. Europhys. Lett. 64, 420-426 (2003). 

This expression was derived by Bell (1978), who used Kramers theory to show that bond lifetime ean be shortened by an applied force in processes such as cell adhesion. Although Eq. (3.2) is quite useful, it is in practice limited, most notably by the fact that it assumes that xp is constant. Typically, measurements of force dependency are made under conditions in which force changes with time, and it is likely that the position of the transition state will move as the shape of the potential surface is perturbed by an applied force (Evans and Ritchie 1997 Hummer and Szabo 2003). Theoretical and empirical treatments of various cases have been put forth in the hterature, but they are outside the scope of this chapter and will not be reviewed here. 

The second approach starts from the modified Langevin equation Eq. (37) and uses the equivalence of the Kramers theory to the multi-dimensional TST. It has been established by numerical comparison that there is agreement between the two approaches. 

E. Poliak The RRKM theory and Kramers theory and its later generalizations by Grote, Hynes, and other are two sides of the same coin. In the spatial diffusion limit, one can show that Kramers s rate expression is identical in form to the RRKM expression, that is, a ratio of equilibrium unidirectional flux and density of reactants. The difficult problem in the application of RRKM theory to the stilbene molecule with a few attached benzenes is whether the equilibration of energy occurs fully on the time scale of the isomerization. One should also... 

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers theory, and in contrast to the inertial motion in Eyring s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or diffusion) in a rod of variable cross-section or in a stream of variable velocity. The magnitude of the diffusion coefficient is established by comparison with the macroscopic kinetics of the change of nuclei, which does not consider fluctuations (cf. Einstein s application of Stokes law to diffusion). The steady rate of nucleus formation is calculated (the number of nuclei per cubic centimeter per second for a given supersaturation). For condensation of a vapor, the results do not differ from those of Volmer. 

Many experiments (see Refs. 154-160) have shown that Kramers theory fails to describe the viscosity dependence of rate in isomerization reactions. This is especially the case where the barrier frequency (< , ) giving the curvature at the barrier top is large. In this case both experimental (see Refs. 154-160) and simulation studies [147, 153] find a rate that decreases with viscosity at a rate much slower than that predicted by Kramers theory. In fact, at high viscosities, it is often found that the rate could be fitted to a form given by... 

In addition, recent investigations should be mentioned that deal with the breakdown of the Kramers theory as a problem of correct modeling232 and its limited ability to describe photophysical processes.233... 

In Kramers theory that is based on the Langevin equation with a constant time-independent friction constant, it is found that the rate constant may be written as a product of the result from conventional transition-state theory and a transmission factor. This factor depends on the ratio of the solvent friction (proportional to the solvent viscosity) and the curvature of the potential surface at the transition state. In the high friction limit the transmission factor goes toward zero, and in the low friction limit the transmission factor goes toward one. 

Since probabilistic dynamics is central to an understanding of Kramers theory for the influence of solvents on the rate constant, we shall first summarize some of the essential features in such a description. 

Kramers theory is based on the Fokker-Planck equation for the position and velocity of a particle. The Fokker-Planck equation is based on the concept of a Markov process and in its generic form it contains no specific information about any particular process. In the case of Brownian motion, where it is sometimes simply called the Kramers equation, it takes the form... 

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