Annihilation rate distribution

Positron annihilation rate distributions for a typical immiscible polymer blend PVC/PS of (80/20) composition and pure PVC and PS are shown in Figure 27.4a, and the radius and free volume cell size distribution PDFs are shown in Figure 27.4b. These PDFs are obtained from CONTIN-PALS2 analysis of the measured lifetime spectra. It is clear from the distribution curves that the width of the curve for PVC is wider compared to PS, while the blend has a distribution in between the constituent polymer distribution, thus demonstrating the ability of the computer routine CONTIN-PALS2 (from the present authors studies). 

Figure 27.4 (a) Positron annihilation rate distribution PDF for PVC and PS polymers and their blend PVC/PS (80/20) composition. Three-peak solution obtained from CONTIN-2 program is plotted (authors studies) ... 

In practice, the positron does influence the charge distribution in the atom or molecule, in such a way as to enhance the electron density in its vicinity. Allowance for this can be made by replacing Z by an effective number of electrons, Thus, the annihilation rate may be expressed as... 

After a sufficiently long time the positrons reach equilibrium, so that y(v,t) = f(v) exp(—(Af)t), where (Af) is the equilibrium annihilation rate and f(v) is the associated speed distribution, which is the solution of the time-independent equation... 

The analyses described above can be applied directly to the equilibrium region of a lifetime spectrum. However, in atomic gases, where slowing down below the positronium formation threshold is by elastic collisions only, the positron speed distribution y(v, t) varies relatively slowly with time. Consequently the annihilation rate also varies slowly with time. From Figures 6.5(a) and (b) the existence of a non-exponential, or so-called shoulder, region close to t = 0 is evident, and the analysis of this region must be treated separately, as outlined below. Further details of the shape and length of the shoulder can be found in subsection 6.3.1 below. 

The x annihilation rate in a DM halo is R = nx(r)(aV)A, where nx(r) = nx,og(r) is the neutralino number density and (cross section averaged over a thermal velocity distribution at freeze-out temperature (see, e.g., Gondolo in these Proceedings). Although the annihilation cross section is a non-trivial function of the mass and physical composition of the neutralino, to our purpose it suffices to recall that the relic density is approximately given by ... 

The most interesting quantities of mixed systems that can be both calculated and measured today are binding energies, annihilation rates, and momentum distributions of the annihilation photons. A system has a binding energy in some state if it is chemically stable in that state, meaning that it stays in that state until it annihilates. More properly, we say that a system is chemically stable in some state if its annihilation rate is greater than the sum of the rates of all other processes that depopulate that state. 

The QMC method is ideally suited for mixed systems because electron-positron correlation, which is difficult to treat with Cl methods, is automatically treated correctly. Systems of up to a bit more than ten leptons are routinely treated. Effective core potential methods can be used to extend the method to larger systems. Expectation values of local operators for the distribution k 2 are calculated by straightforward sampling procedures, but nonlocal operators, such as those for the annihilation rate, are problematic and are under active investigation [12],... 

As can be seen in Figure 7.15 very different lifetimes occur and each dataset shows different lifetimes. In principle the sample material and the size distribution of pores and the open porosity component each generate an annihilation rate (f ) with some relative intensity (I ). The spectrum is then convoluted with the experimental response function (R). Random statistical noise and a certain background (B) level are added. The measured time spectmm M(t)... 

The lifetime distribution for PTMSP at ambient ten rature is shown in Figure 1. Using this distribution, the size distribution of FV was calculated by means of Equation 1. Naturally, the o-Ps size distribution consists of two peaks (Figure 2). The dashed line in Figure 2 shows the calculated dependence of the annihilation rate (Xi=l/Xi ns ) versus FV radius used in conq>uting the size distribution by means of Equation 1. 

Figure 2. Free-volume hole radius distribution R pdf(R) (relative units) of PTMSP (pdf(R) is probability density function) obtained from the two right-hand peaks in Fig. 1. The dashed line shows the calculated dependence (3) of annihilation rate, (in ns ) versus radius of FV elements used in confuting the size distribution.

Since the latter conditions pertain to aromatic nitration solely via the homolytic annihilation of the cation radical in Scheme 16, it follows from the isomeric distributions in (81) that the electrophilic nitrations of the less reactive aromatic donors (toluene, mesitylene, anisole, etc.) also proceed via Scheme 19. If so, why do the electrophilic and charge-transfer pathways diverge when the less reactive aromatic donors are treated with other /V-nitropyridinium reagents, particularly those derived from the electron-rich MeOPy and MePy The conundrum is cleanly resolved in Fig. 17, which shows the rate of homolytic annihilation of aromatic cation radicals by NO, (k2) to be singularly insensitive to cation-radical stability, as evaluated by x. By contrast, the rate of nucleophilic annihilation of ArH+- by pyridine (k2) shows a distinctive downward trend decreasing monotonically from toluene cation radical to anthracene cation radical. Indeed, the... 

In simulations [9] Sierpinski gaskets on the 12th stage, containing 177147 or 265722 sites, were used respectively. The number No of randomly distributed A or B particles was 10 percent of the total number of sites. The random mutual annihilation of dissimilar particles was simulated through a minimal process method [10] from all AB pairs at each reaction step one pair was selected randomly, according to its reaction rate (3.1.2) the time... 

Here p is a defect creation rate per unit time and volume, called also dose rate, f r) is their initial distribution function over relative distances, normalised according to f f(r)dr= 1, o(r) the AB pair recombination rate. For the annihilation mechanism o(r) = cro0(ro - r), 0 is the Heaviside step-function (Section 3.1). 

Here ct, ce and cp are the concentrations of the positive ions, electrons and the positron probability density at a point r measured from the center of the blob at time t. Dp is the diffusion coefficient of the positron, Di = De = Damb 0 is the ambipolar diffusion coefficient of the blob, a2 o2 ss a2 is the dispersion of the intrablob species, and a2 is the dispersion of the positron space distribution by the end of its thermalization. Decay rate Te-1 = 1/t + kescs is the sum of the electron solvation rate and possible capture by solute molecules t 2 = 1 /t2 + l/r + kpscs accounts for the free e+ annihilation, solvation and reaction with S. Similarly, t 1 = l/rjmr + hscs, where T r is the rate of the ion-molecule reaction. 

In a 6-sq.-mile test of the male annihilation technique initiated in 1952 in Hawaii the methyleugenol, which oriental fruit flies consume avidly, was exposed on sugarcane flberboard squares, distributed at the rate of about 40 per sq. mile. Pyrolan (3-methyl-l-phenyl-5-pyrazolyl dimethyl carbamate), a strong stomach poison, was added to the lure to kill the male flies. The poisoned feeding stations were retreated each month. Reductions in infestation averaged 74, 70, 82, and 60% at 700-, 1100-, 1500-, and 1900-foot elevations in the treated area, again with only partial isolation. Infestations rapidly increased after termination of the experiment. 

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