Positron rate distribution

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. 

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. 

As 73 has a small intensity, it is subtracted fiem the spectra and the two-state trapping model is used to determine the bulk lifetime Tb, according to eq. (20). The compilation of the results found in the hterature gives Tb = 184 11 ps and 215 4ps for the metaUic and insulating phase, respectively. These values are statistically different, contrary to tm, T] and t2- The interpretation of this difference can be based on the calculated aimihilation rates which depend on the positron density distribution in the lattice unit cell. 

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... 

As outlined by Wolf (1993) and Holzscheiter et al. (1996), similar considerations to those described above also apply to recombination in traps, and in particular the nested Penning-trap scheme (see below), again with appropriate assumptions regarding the speed distributions of the trapped positrons and antiprotons and their degree of spatial overlap. As an example, Holzscheiter et al. (1996) argued that the recombination rates are of the order of one per second (though dependent upon Ee) for 106 positrons and 105 antiprotons trapped in a volume of 1 cm3. 

The wavelength dependence of laser-stimulated recombination could be used to perform spectroscopy. The recombination rate is closely related to the distribution of positrons in energy and also to the population of bound levels close to the ionization threshold [17]. The spectral resolution for the first step of laser-stimulated recombination is thus limited by the energy distribution of the positrons. Laser-induced two-step recombination, first into a high-lying state with the subsequent stimulation of a bound-bound transition into a lower lying state, offers a first possibility for precise laser spectroscopy [18]. 

Amines labelled with positron-emitting nuclides (11C, 13N, 18F) combined with PET are useful for external measurement of the rate of transport, metabolism and excretion of a number of substances in humans and animals, and were used for various aims. After a detailed study of the organ distribution and the metabolic fate of the 13N-amines, 13N-/ -phenethylamine, 13N- -octylamine and 13N-3,4-dimethoxyphenethylamine, a general transformation path of amines in vivo has been formulated292 (equation 130). 

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],... 

Now the field is turned on. Using the coordinate system presented in fig. 8, we have Sy, II (-z) and By, x (since By is assumed to be the external field only). It follows that Sy processes in the ( v,z)-plane. The angular distribution W(0) of the emitted positrons is coupled to the motion of At time zero, Sy points away from Dp and Nf (t) oc (1-ao). Half a precession period later, Sy points towards Dp and Nf(t) oc (1 -I- ao). The count rate Nf(t) will thus be modulated with the amplitude ao and the muon spin precession frequency fy. One finds... 

The most interesting information from the chemical point of view may be obtained by measurement of the distribution of the lifetimes of the positrons or positronium atoms. In addition to the lifetime, this method also gives a measure of the relative probability of ort/io-positronium formation as a separable parameter. Hence, from the lifetime distribution, information is obtained not only on the rates of the chemical reactions and other interactions of the positronium, but also on the probability of positronium formation, e.g., the extents of the possible inhibitory processes. The only deficiency of the method is that it does not provide a possibility for simple differentiation between ortho-para conversion processes and the chemical reactions, since both processes cause a decrease in the lifetime. 

Positron annihilation lifetime spectroscopy (PALS) is an efficient tool for measuring free volume and sizes of free volume elements in polymeric materials. This is particularly inq)ortant for studies of membrane materials, since free volume determines the permeation rate of small molecules. Free volume was studied by means of PALS in polymers characterized by extremely high permeability poly(l-trimethylsilyl-l-propyne) and copolymers of 2,2-bistrifluoromethyl-4,5-difluoro-l,3-dioxole and tetrafluoroethylene. The results obtained were compared with those observed for conventional glassy polymers. For the first time, the size distribution of free volume has been determined for these membrane materials. 

The simple trapping model (STM) can be used to interpret the results from PL measurements [129], According to this model, for positrons implanted into a homogeneous solid with a bulk lifetime z, = 1/Ab and with N different kinds of homogeneously distributed microscopic defects with lifetimes Xi = jA and trapping rates ki, the annihilation spectrum will consist of A +1 exponentials. Through defect trapping, the bulk lifetime will be reduced to tq = l/(Ab -I- k), where... 

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