Annihilation rates

Assume the edge dislocation density to be divided into positive and negative populations, N+ and N, moving only on slip planes at 45° (maximum shear stress) to the planar shock front. For a dislocation multiplication (annihilation) rate M, show that conservation of dislocations requires that... 

Characteristics of PPLEDs with PVK (4)—Ir Complex (5 wt%) Blends External QE (r/exi) and Power Efficiency (i7p) Are Given as Functions of Current Density Amax Is the EL Emission Peak the J0 Is the Current at Which the rjext Falls to 50% of Its Peak Value and Is Characteristic of a Triplet-Triplet Annihilation Rate)... 

An important characteristic of positronic systems relevant to the experiment is their lifetimes. The expectation value of the electron-positron contact density allows us to evaluate the two-photon annihilation rate for a positronic system using the expression... 

It is interesting to note that excimer bands of phenanthrene67 and of anthracene,124 which have defied detection in the prompt fluorescence spectra even at low temperatures, have been observed in the delayed emission spectra of these compounds at — 75°K. Presumably at the low temperatures necessary to observe these bands the high solvent viscosity completely suppresses photoassociation at the reduced concentration available, i.e., WM] 1 /r , whereas the reduced triplet-triplet annihilation rate constant mstationary concentration of the triplet state. 

Let us analyze these results one step further and ask about a quantitative measure of the Kirkendall effect. This effect had been detected by placing inert markers in the interdiffusion zone. Thus, the lattice shift was believed to be observable for an external observer. If we assume that Vm does not depend on concentration and local defect equilibrium is established, the lattice site number density remains constant during interdiffusion. Let us designate rv as the production (annihilation) rate of the vacancies. We can derive from cA+cB+cv = l/Vm and jA +/ B +./v = 0 that... 

An important feature of o-Ps in polymers is that these particles tire preferentially formed or trapped in holes or regions of low electron density. The annihilation rate of o-Ps is proportional to the overlap of the positron and the pick-off electron wavefunctions and therefore the lifetime of o-Ps will depend on the size of the hole. The relative number of o-Ps pick-off annihilations is related to the number of suitable free volume sites in the polymer [3]. 

Note that <727 — 00 as v — 0, although the annihilation rate, which is proportional to the product W27, remains finite. At low incident positron energies the two gamma-rays are emitted almost collinearly, the energy of each being close to me2 (= 511 keV). Annihilation of a small fraction of the positrons emanating from the radioactive source can occur at relativistic speeds and then it is necessary to use the full equation (1.2). 

The lowest order contributions to the annihilation rates for the nPs1So and nPS3Si states of positronium were first calculated by Pirenne (1946)... 

If the electrons are bound in atoms or molecules, each having Z electrons, and the number density of atoms or molecules is n, the electron density is ne = nZ. Therefore, if there were no distortions of the positron-atom system, the free annihilation rate would be given by... 

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... 

In cases where the annihilation rate, and hence Zef[, is a rapidly varying function of the positron energy, as with xenon (Schrader and Svetic, 1982), the simplification introduced above is not valid and the solution to equation (6.15) must be used. The functional form for f(v) given in equation (6.17) was used by Campeanu and Humberston (1977b) to investigate the variation of the equilibrium value of (Zeff) with electric field and temperature, and their results for the former are shown in Figure 6.3. 

Once the background is subtracted, the component of the spectrum due to the annihilation of ortho-positronium is usually visible (see Figure 6.5(a), curve (ii) and the fitted line (iv)). The analysis of the spectrum can now proceed, and a number of different methods have been applied to derive annihilation rates and the amplitudes of the various components. One method, introduced by Orth, Falk and Jones (1968), applies a maximum-likelihood technique to fit a double exponential function to the free-positron and ortho-positronium components (where applicable). Alternatively, the fits to the components can be made individually, if their decay rates are sufficiently well separated, by fitting to the longest component (usually ortho-positronium) first and then subtracting this from the... 

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 instantaneous free-positron annihilation rate is particularly useful where it is time dependent (i.e. on the shoulder region) it is... 

In this section we review the results from positron annihilation experiments, predominantly those performed using the lifetime and positron trap techniques described in section 6.2. Comparisons are made with theory where possible. The discussion includes positron thermalization phenomena and equilibrium annihilation rates, and the associated values of (Zeff), over a wide range of gas densities and temperatures. Some studies of positron behaviour in gases under the influence of applied electric fields are also summarized, though the extraction of drift parameters (e.g. mobilities) is treated separately in section 6.4. Positronium formation fractions in dense media were described in section 4.8. 

Measurements of the equilibrium annihilation-rate parameter, (Zeff), have been made for a plethora of gases over a wide range of densities. In this section we confine ourselves to the low density and high temperature region, in which many-body effects can be ignored and where the results may be compared with those obtained from scattering theory, as outlined in section 6.1. 

The positron-trap technique has been used to measure the annihilation rate of positrons interacting with a wide variety of molecules. The species investigated by Iwata et al. (1995) include many hydrocarbons, substituted (e.g. fluorinated and chlorinated) hydrocarbons and aromatics as mentioned in section 6.1, large values of (Zeff) (in excess of 106) were found for some molecules. Several distinct trends are exhibited in the data of Iwata et al. (1995). Though much of the detailed physics involved in the annihilation process on these large molecules is still unclear, the model of Laricchia and Wilkin (1997), described in section 6.1, may offer a qualitative explanation of the observations. 

The temperature, or energy, dependence of the annihilation rate, or (Zeflf), has also been investigated using a positron trap. In this technique positrons are first accumulated at room temperature, and then their... 

Fig. 6.9. Positron annihilation rates at various temperatures (normalized to unity at room temperature) for the noble gases, from Kurz et al. (1996). Key o, He , Ne . Ar A, Kr , Xe. The curves are from the theoretical work of...

Kurtz, Greaves and Surko, Temperature dependence of positron annihilation rates in noble gases, 2929-2932, copyright 1996, by the American Physical Society. 

Fig. 6.10. Mean positron annihilation rate (denoted here as Af) at various gas densities for N2 and Ar gases at different temperatures. Key A, N2 at 130 K , Ar at 160 K A, N2 at 297 K , Ar at 297 K. The original sources for these measurements are given by Heyland et al. (1986). The broken line indicates the linear rise expected, equation (6.3), for a constant (Zeff) of 27. Reprinted from Physics Letters A119, Heyland et al., On the annihilation rate of thermalized free positrons in gases, 289-292, copyright 1986, with permission from Elsevier Science.

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