Electron trajectories, Monte-Carlo

An important difference between the wave treatment and the SCA is that energy conservation is retained exactly and when the energy of the projectile is less than the energy required to excite the state under consideration the cross section is zero. This is called a threshold. In fig. 5.8 is plotted the measured ratio for ionization produced by equal-velocity positron and proton projectiles incident on helium. Just above the threshold, 24eV, Uie electron cross section falls below that due to the heavier proton. The ratio is compared to a CTMC (classical trajectory Monte Carlo) prediction and also to the ratio of the PWBA to the SCA cross sections showing the importance of the mass of the projectile to the result. The CTMC method will be discussed in more detail shortly. 

Kuntz, P. J. and Schmidt, W. E, A classical trajectory Monte Carlo model for the injection of electrons into gaseous argon, /. Chem. Phys., 76,1136,1982. 

Monte Carlo electron trajectory simulations provide a pictorial view of the complei electron—specimen interaction. As shown in Figure 2a, which depicts the interac-... 

Figure 2 (a) Monte Carlo simulation of electron trajectories in copper beam energy... 

The phenomena of beam broadening as a function of specimen thickness are illustrated in Fig. 4.20 each figure represents 200 electron trajectories in silicon calculated by Monte Carlo simulations [4.91, 4.95-4.97] for 100-keV primary energy, where an infinitesimally small electron probe is assumed to enter the surface. In massive Si the electrons suffer a large number of elastic and inelastic interactions during their paths through the material, until they are finally completely stopped. The resulting penetration depth of the electrons is approximately 50 pm and in the... 

The MD/QM methodology [18] is likely the simplest approach for explicit consideration of quantum effects, and is related to the combination of classical Monte Carlo sampling with quantum mechanics used previously by Coutinho et al. [27] for the treatment of solvent effects in electronic spectra, but with the variation that the MD/QM method applies QM calculations to frames extracted from a classical MD trajectory according to their relative weights. 

Fig. 10. The emerging picture of electronically nonadiabatic interactions of NO molecule scattering at a metal surfaces. Transition from the ground electronic state to an anionic state which is strongly attractive to the metal surface can be accomplished by high translational energy when vibrational excitation is low (black trajectory). When vibrational motion is highly excited, even low translational energies allow transition of the anionic state (red trajectory). Recently, Monte-Carlo wavepacket calculations have been carried out which tend to support this picture.63...

Figure 27. Monte Carlo simulated trajectories of 100 electrons in PMMA resist on silicon (Reproduced with permission from Ref. 42)...

These experiments are important because they are performed on a reaction for which a priori calculations of V(rAB, rBC, rCA) are likely to have their best chance of success as only three electrons are involved. Even here the accurate computation of V, frequently termed the potential-energy hypersurface, is extremely difficult. Porter and Karplus [19] have determined a semiempirical hypersurface, and Karplus, Porter, and Sharma [20] have calculated classical trajectories across it. This type of computer experiment has been mentioned before and will be described in greater detail later. The objective of Karplus et al. was to calculate aR(E) and E0. Collisions were therefore simulated at selected values of E, with other collision parameters selected by Monte Carlo procedures, and the subsequent trajectories were calculated using the classical equations of motion. Above E0, oR was found to rise to a maximum value, of the same order of magnitude as the gas-kinetic cross section, and then gradually to decrease to greater energies. 

Monte Carlo simulations performed by the author using the program CASINO ( monte CArlo Simulation of electroN trajectory in sOlids ), available free-of-charge on the Internet http //www.gel.usherbrooke.ca/casino/What.html... 

It is also important to know that the interaction volume of electrons with a solid specimen is much larger than the beam size. According to a Monte Carlo simulation, in which the detailed history of an electron trajectory is calculated in a stepwise manner, the interaction volume is a function of the accelerating voltage and properties of the target specimen. 

The energy of the electron beam is also an important factor for the time resolution because of the scattering of the electron beam in irradiated samples (usually condensed media). Eigure 4 shows the electron penetration into liquid water and trajectories calculated by a Monte Carlo calculation code (EGS5 ) for electron penetration into... 

Figure 4.9 Monte Carlo electron trajectory simulation of an electron beam interaction with iron E = 20 keV. (Reproduced with kind permission of Springer Science and Business Media from J.I. Goldstein et al, Scanning Electron Microscopy and X-ray Microanalysis, 2nd ed., Plenum Press, New York. 1992 Springer Science.)...

The time evolution of the electronic wave function can be obtained in the adiabatic or in the diabatic basis set. At each time step, one evaluates the transition probabilities between electronic states and decides whether to hop to another siu-face. When hopping occurs, nuclear velocities have to be adjusted to keep the total energy constant. After hopping, the forces are calculated from the potential of the newly populated electronic state. To decide whether or not to hop, a Monte Carlo technique is used Once the transition probability is obtained, a random number in the range (0,1) is generated and compared with the transition probability. If the munber is less than the probability, a hop occurs otherwise, the nuclear motion continues on the same surface as before. At the end of the simulation, one can analyze populations, distribution of nuclear geometries, reaction times, and other observables as an average over all the trajectories. 

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