Energy transmission probability

A and A = 0.1 eV. The adiabatic ground potential energy surface is shown in Fig. 11. The present results (solid line) are in good agreement with the quantum mechanical ones (solid circles). The minimum energy crossing point (MECP) is conventionally used as the transition state and the transition probability is represented by the value at this point. This is called the MECP approximation and does not work well, as seen in Fig. 10. This means that the coordinate dependence of the nonadiabatic transmission probability on the seam surface is important and should be taken into account as is done explicitly in Eq. (18). 

Zhu and Nakamura proved that the intriguing phenomenon of complete reflection occurs in the ID NT type potential curve crossing [1, 14]. At certain discrete energies higher than the bottom of the upper adiabatic potential, the particle cannot transmit through the potential from right to left or vice versa. The overall transmission probability P (see Fig. 45) is given by... 

Figure 62. Hydrogen transmission probability as a function of total energy (eV) for the center approach. Taken from Ref [47].

In this energy region, physically meaningful quantities are the overall transmission and reflection probabilities. The transmission probability is given by... 

Fig. 6.2 (a) Bell (parabolic) and Eckart barriers, both widely used in approximate TST calculations of quantum mechanical tunneling, (b) Transmission probability (Bell tunneling) as a function of energy for two values of the reduced barrier width, a... 

The tunnelling correction P is the transmission probability through the potential barrier averaged over all possible crossing points and potential energies . An asymmetrical banier of the Eckart type l is assumed in the present model. 

Figure 6 shows the results obtained for N(E) for several values of A. We do not obtain satisfactory results for A = 0, but for a wide range of A > 0 we obtain quite stable results that are relatively insensitive to the particular value of this smoothing parameter. This is precisely the behavior one wishes to see. It is also significant that the results in Fig. 6 are accurate for some ways into the classically forbidden tunneling regime, in this case for energies as much as 0.1 eV or so below the barrier, down to a transmission probability of 10"3. 

There are two main ingredients that go into the semiclassical turnover theory, which differ from the classical limit.51 In the latter case, a particle which has energy E > 0 crosses the barrier while if the energy is lower it is reflected. In a semiclassical theory, at any energy E there is a transmission probability T(E) for the particle to be transmitted through the barrier. The second difference is that the bath, which is harmonic, may be treated as a quantum mechanical bath. Within first order perturbation theory, the equations of motion for the bath are those of a forced oscillator, and so their formally exact quantum solution is known. 

The inelastic transmission matrix T(e, e) describes the probability that an electron with energy e, incident from one lead, is transmitted with the energy e into a second lead. The transmission function can be defined as the total transmission probability... 

The importance of the two dimensional periodicity on the transmission properties is demonstrated in Figure 5, which presents the transmission probability of electrons as a function of the photoelectron energy for layers of Cdar (dashed), Cdbr (dotted) and of mixed layers (solid) for three (Fig. 5A) and nine (Fig. 5B) layers. As is clearly evident, the electron transmission through the mixed layers is significantly less efficient than that through the Cdar or Cdbr layers themselves. Moreover, the spectmm for the mixed layers is much closer to the relaxed type (Fig. 2). 

If we send in a particle (say, from the left) with kinetic energy E, then it can be shown that the probability of crossing the barrier (i.e., the transmission probability)... 

In a classical calculation, the transmission probability will therefore be underestimated at kinetic energies smaller than the barrier energy and overestimated at kinetic energies above the barrier energy... 

Note that this result is identical to Eq. (6.24). Although the quantum mechanical transmission probability at kinetic energies above the barrier energy is less than one, that is, particles are reflected above the barrier, the transmission in the tunneling region dominates in the integral due to the Boltzmann factor exp(—E/kBT). 

Fig. 9.9. Transmission probability (T79 of the Csl molecule as a function of the total energy E. (a) Irregular fluctuations in the classically chaotic regime (Vb > (b) Smooth behaviour in the classically regular regime (Vb (Adapted from Bliimel and Smilansky (1988).)...

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