Projected range calculation

This formula gives values correct to about 15%, but a more exact relation between range and projected range has been calculated using a power law-based LSS theory (11). 

An estimate of AR ia alloys can be made usiug the empirical expression (12) of equation 10 where the average alloy reduced energy, is defined by equation 11, where C (i = 1, 2,... , n) is the elemental atomic fraction of the /th element, and is the elemental reduced energy defined iu equation 3. Using this formulation, the projected range straggling iu compounds can be calculated to within 20%. 

Figure 4.36. Comparison of experimental results with calculated values of projected ranges (i p) as a function of implant energy for P in Si. (Reproduced by permission of Kobayshi and Gibson 1999.)...

Fig. 5.1. Opacity of stellar material with X = 0.7, Z = 0.02 (roughly solar composition) as a function of temperature and the parameter log R, where R = p/T is approximately constant throughout a main-sequence star, corresponding to a polytrope with n = 3 (see Appendix 4) e.g. at 1 M0 log/ varies from —1.5 at the centre to 0.0 in the envelope, while at 10 M0 the corresponding range is from -3.5 to -4.0. (Density in units of gmcm-3, T(, in units of 106K, opacity in units of cm2 gm-1.) OP and OPAL refer to two independent opacity calculation projects. After Badnell, Bautista, Butler et al. (2005).

The diffusional displacement of B is a function of implant dose and energy. The energy dependence is illustrated in Figure 24, which shows the diffusion of B at a concentration of 1 X 1017/cm3 versus Rp, the projected range of B implantation. The implants were 1 X 1014-2 X 1014 B atoms per cm2 annealed at 800-850 °C for approximately 0.5 h. The displacement increases with implant depth and then reaches saturation. The calculated curve in Figure 24 is based on the concentration of excess self-interstitials in the tail of the implant that increases directly with range, up to a maximum value. 

Biersack, J.P. Calculation of projected ranges in analytical solutions and a simple general algorithm. Nucl. Instrum. Meth. 182/183, 199 (1981)... 

Fig. 11. Mean projected range vs. incident energy for He" incident on silicon. The experimental results ( ) are compared with mean ranges obtained from analytical theory (solid line) and from TRIM calculations (dashed line) (after Eckstein, 1991).

Figure 2 shows the reference coordinates and nomenclature of this geometry. Theoretical calculations of the projected range and straggle for various dopants and substrates have been calculated and shown in graphical form (4, 21-26). Some common dopant ions for silicon and gallium arsenide are shown in Figures 3, 4, 5, and 6. 

Figure 3. Theoretical calculations of the projected range of B, P, As, and Sb in silicon. (Adapted with permission from Reference 27, copyright 1983, John Wiley and Sons).

FIGURE 7 Boron depth profiles for polyethylene (PE) and polyamide 6 (PA) implanted with 5 X 10 B /cm at 100 keV. Theoretical depth profile for boron implanted in polyethylene is shown by the histogram (calculations were performed with the TRIM code is the theoretical projected range). 

Computer simulation programs have been developed in order to calculate and investigate the structure of cascades and the projected ranges of ions in solids [78-81]. These calculations are based on the assumptions that the collisions are of a two-body type and that energetic atoms collide only with stationary ones. The most popular program is the Monte Carlo code named TRIM (transport of ions in matter) [82J. This program does not take into account changes such as displacements and phase formation or the amorphization in-... 

Stepwise creation of Gaussian implantation profiles, where the projected ranges are calculated with the stopping powers given by Kalbitzer [85] and convoluted with a width due to straggling. 

The numerical values of and a, for a particular sample, which will depend on the kind of linear dimension chosen, cannot be calculated a priori except in the very simplest of cases. In practice one nearly always has to be satisfied with an approximate estimate of their values. For this purpose X is best taken as the mean projected diameter d, i.e. the diameter of a circle having the same area as the projected image of the particle, when viewed in a direction normal to the plane of greatest stability is determined microscopically, and it includes no contributions from the thickness of the particle, i.e. from the dimension normal to the plane of greatest stability. For perfect cubes and spheres, the value of the ratio x,/a ( = K, say) is of course equal to 6. For sand. Fair and Hatch found, with rounded particles 6T, with worn particles 6-4, and with sharp particles 7-7. For crushed quartz, Cartwright reports values of K ranging from 14 to 18, but since the specific surface was determined by nitrogen adsorption (p. 61) some internal surface was probably included. f... 

Many project managers find it realistic to estimate time intervals as a range rather than as a precise amount. Another way to deal with the lack of precision in estimating time is to use a commonly accepted formula for that task. Or, if you are working with a mathematical model, you can determine the probability of the work being completed within the estimated time by calculating a standard deviation of the time estimate. 

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