Monte Carlo simulation electron trajectories

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... 

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

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. 

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

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. 

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. 

Figure 3 Monte Carlo electron trajectory simulation for four different materials. Beam energy, 25keV beam diameter, 1 nm tilt, 0° 150 trajectories. Targets (density (gcm ) (A) carbon (2.26) (B) silicon (2.33) (C) silver (10.50) and (D) gold (19.28).

Molecular simulations Computer modeling of the motion of an assembly of atoms or molecules. In molecular simulations only the motions of nuclei are considered, (i.e one assumes that the electronic Schrodinger equation has been solved providing intermolecular interaction potentials.) In practice empirical interaction potentials are utilized in most cases. Two main approaches are used the Monte Carlo (MC) method and molecular dynamics (MD). The former relies on statistical sampling of the configuration space of the systems, whereas the latter solves classical mechanics (Newton) equations to find trajectories of molecules. 

Fig. 6.3 Simulation of the interaction between incident electrons and the sample, as a function of incident beam voltage, by Monte Carlo calculation of the electron trajectories [49]. The volume is much less at low beam voltages. (From Joy and Pawley [49] reproduced with permission.)...

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