Path length distribution

Thus the spatial rate of energy loss is (q)/A. The stopping power is actually the mean value of the ratio of energy loss to path length, and to this extent the derivation is an approximation. The path length distribution Pjx) in n collisions may be given as a convolution—that is,... 

P]j(0, i — a), Vi Z and k K. In our example, when the particle displacement is realized by unitary steps and in a positive direction (type I) or in a negative direction (type II) and where the path length distribution is uniform with a unitary value for both component processes, we have ... 

The path length distribution of a network tells us how far nodes are from each other. Scale-free networks are ultra-small because they have an average path length of the order log(log N), where N is the number of nodes. Random networks are small because their mean path length is of the order logN. ... 

Discrimination by adjustable path length in gas-flow Geiger Counters, 258 Discriminators, see Pulse-height discriminator Pulse-height selection Distribution of sample in spot test, 228 Divergence of x-ray beam in slit system, 111, 112... 

This is an indication of the collective nature of the effect. Although collisions between hard spheres are instantaneous the model itself is not binary. Very careful analysis of the free-path distribution has been undertaken in an excellent old work [74], It showed quite definite although small deviations from Poissonian statistics not only in solids, but also in a liquid hard-sphere system. The mean free-path X is used as a scaling length to make a dimensionless free-path distribution, Xp, as a function of a free-path length r/X. In the zero-density limit this is an ideal exponential function (Ap)o- In a one-dimensional system this is an exact result, i.e., Xp/(Xp)0 = 1 at any density. In two dimensions the dense-fluid scaled free-path distributions agree quite well with each other, but not so well with the zero-density scaled distribution, which is represented by a horizontal line (Fig. 1.21(a)). The maximum deviation is about... 

Fig. 1.21. Ratio of free-path distribution Xp to the scaled zero-density free-path distribution plotted as a function of reduced free-path length r/X for two-dimensional (a) and three-dimensional (b) liquids. Circles, inverted triangles and upright triangles refer to reduced volumes V/V0 of 1.6, 2, and 3, respectively (V0 is the volume of the system at close packing) [74].

Specific solar radiation conditions are defined by the air mass (AM) value. The spectral distribution and total flux of radiation outside the Earth s atmosphere, similar to the radiation of a black body of 5,900 K, has been defined as AM-0. The AM-1 and AM-1.5 are defined as the path length of the solar light relative to a vertical position of the Sun above the terrestrial absorber, which is at the equator when the incidence of sunlight is vertical (90°) and 41.8°, respectively. The AM-1.5 conditions are achieved when the solar flux is 982 Wm2. However, for convenience purpose the flux of the standardized AM-1.5 spectrum has been corrected to 1,000 Wm2. 

In Table 5.1, the theoretical effective absorption path length is calculated from (5.5) using, for example, in the case of methane, x = 0.28 (calculated from the dip depth M0 the dip gets shallower with analyte absorption, so the WGM is undercoupled), / = 1.6% (estimated from a computation of the field distributions at the same wavelength in a microsphere of the same diameter), and oq = 0.0061 cm 1... 

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