Positron diffusion

Additional, but rather less direct, evidence for the accuracy of the variational results for models H5 and H14 is provided by the excellent agreement between the theoretical and experimental lifetime spectra for positrons diffusing in helium gas, where calculation of the theoretical spectrum requires a knowledge of the momentum transfer and annihilation cross sections, both of which are derived from the wave functions generated in the calculations of the elastic scattering phase shifts. A detailed discussion of positron lifetime spectra is given in Chapter 6. 

Traditionally, experimental values of Zeff have been derived from measurements of the lifetime spectra of positrons that are diffusing, and eventually annihilating, in a gas. The lifetime of each positron is measured separately, and these individual pieces of data are accumulated to form the lifetime spectrum. (The positron-trap technique, to be described in subsection 6.2.2, uses a different approach.) An alternative but equivalent procedure, which is adopted in electron diffusion studies and also in the theoretical treatment of positron diffusion, is to consider the injection of a swarm of positrons into the gas at a given time and then to investigate the time dependence of the speed distribution, as the positrons thermalize and annihilate, by solving the appropriate diffusion equation. The experimentally measured Zeg, termed Z ), is the average over the speed distribution of the positrons, y(v,t), where y(v,t) dv is the number density of positrons with speeds in the interval v to v + dv at time t after the swarm is injected into the gas. The time-dependent speed-averaged Zef[ is therefore... 

Fig. 6.2. The time dependence of (Zeg for positrons diffusing in helium gas at room temperature and zero electric field. Experiment — —, Coleman et al.

The contribution of positron diffusion length (L+ = 10 nm [22]) was removed from the escape depth values. The diffusion constant in a material is a function of diffusion length and annihilation rate D = L2X. Here, the rates for positrons and positronium are similar (X 2 ns 1). Thus the measured combined effective diffusion length of positrons L+ and positronium escape Lesc is l eff = L2+ + L2esc [30],... 

Figure 7.8 Escape depth from which pores are connected to the surface versus porogen load. The contribution of positron diffusion was removed from the fitted diffusion length to obtain the positronium escape depth. At 50% porogen load the escape depth exceeds the thickness of the samples. The present model does not include this and becomes...

Positron lifetime measurements can be used to investigate the type and the density of lattice defects in crystals [293]. In solid materials positrons have a typical lifetime of 300 to 500 ps until they are annihilated by an electron. When positrons diffuse through a crystal they may be trapped in crystal imperfections. The electron density in these locations is different from the density in a defect-free crystal. Therefore, the positron lifetime depends on the type and the density of the crystal defects. When a positron annihilates with an electron two y quanta of 511 keV are emitted. The y quanta can easily be detected by a scintillator and a PMT. 

The trapping of positrons in defects is based on the formation of an attractive potential at open-volume defects, such as vacancies, vacancy agglomerates, and dislocations. The main reason for this potential is the lack of a repulsive positively charged nucleus in such a defect. The sensitivity range for vacancy detection in metals starts at about one vacancy per 10 atoms. This extremely high sensitivity is caused by the fact that the positron diffuses about 100 nm through the lattice and... 

Positrons diffusing through matter can be captured in special trapping sites. As shown in early studies, these trapping centres are crystal imperfections, such as vacancies and dislocations. The wavefunction of a positron captured in such a defect is localised until it annihilates with an electron of its immediate surroundings into y-rays. Since the local electron density and the electron momentum distribution are modified with respect to the defect-free crystal, the annihilation radiation can be utilised to obtain information on the localisation site. The different positron techniques are based on analysing the annihilation radiation. The principles of the basic positron methods are illustrated in Figure 4.27 [84]. 

Positrons may be trapped by grain boundaries (GBs) in metals. Nevertheless, trapping in GBs is likely only when the mean linear dimension of the grains does not exceed a few pm. This means that the grain size is comparable to (or smaller than) the positron diffusion length L+, and some of the positrons have a chance of reaching a GB by diffusive motion. The movement of a positron to a GB substantially limits the positron trapping rate in GBs [91]. 

The semi-classical random walk theory gives the following expression for the positron diffusion coefficient ... 

Positron diffusion coefficient at room temperature. For experimental values ... 

The symbol E+ denotes the total energy of the crystal with the positron in its lowest state (Bloch state at — 0), and v is the crystal volume. In metals, Ed is typically on the order of - p (Ep is the Fermi energy) [73]. Positron diffusion constants calculated using Equation 4.105 are listed in Table 4.19. Some experimental values of D+ that can be found in the literature are also included for comparison. We can see that there is relatively good agreement with the experimental values. 

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