Positron annihilation theory

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. 

I.A. Ivanov, J. Mitroy, Optical model theory for positron annihilation during scattering, J. Phys. B At. Mol. Opt. Phys. 33 (2000) L831. 

Following the theory [20,22], the most probable native defects in GaN crystals which are highly n-type are Ga-vacancies (Vo,3 ). This is due to their low formation energy even under strongly Ga-rich conditions of crystallisation. Hie presence of negatively charged Ga vacancies at concentrations of 1018 cm 3, in n-type pressure grown GaN crystals, was recently detected by positron annihilation experiments... 

The first anti-particle discovered was the anti-electron, the so-called positron, in 1933 by Anderson [3] in the cloud chamber due to cosmic radiation. The existence of the anti-electron (positron) was described by Dirac s hole theory in 1930 [4], The result of positron—electron annihilation was detected in the form of electromagnetic radiation [5]. The number and event of radiation photons is governed by the electrodynamics [6, 7]. The most common annihilation is via two- and three-photon annihilation, which do not require a third body to initiate the process. These are two of the commonly detected types of radiation from positron annihilation in condensed matter. The cross section of three-photon annihilation is much smaller than that of two-photon annihilation, by a factor on the order of the fine structure constant, a [8], The annihilation cross section for two and three photons is greater for the lower energy of the positron—electron pair it varies with the reciprocal of their relative velocity (v). In condensed matter, the positron—electron pair lives for only the order of a few tenths to a few nanoseconds against the annihilation process. 

The aim of this chapter is to introduce the reader to the application of positron annihilation techniques to polymers. An extensive review of the large volume of publications related to positron studies in polymers will not be presented. Rather it is intented to introduce the reader to the theory and techniques used in polymer studies and indicate the types of information that can be obtained about different polymer systems. The main focus of this chapter will be on the use of positron annihilation lifetime spectroscopy (PAL) in polymer studies. Chapter 11 discusses the use of monoenergetic slow positron beams used to study polymers surfaces. One of the interesting new developments in the application of positron annihilation techniques in polymers is the positron age-momentum correlation technique (AMOC). This technique promises to shed new light on the mechanisms of positronium formation and annihilation in polymer systems. A more detailed discussion of this technique can be found elswhere in this text. 

Consolati, G., Quasso, R, Simha, R., and Olson, G. B., On the relation betwen positron annihilation lifetime spectroscopy and lattice-hole-theory free volume, J. Polym. Sci. B, 43, 2225-2229 (2005). 

Liu, J., Deng, Q., and Jean, Y. C., Free-volume distributions of polystyrene probe by positron annihilation comparison with free-volume theories. Macromolecules, 26,7149-7155 (1993). 

Typically, therefore, a PALS spectrum consists of a minimum of three components the short-lived p-Ps component with intensity 7i and lifetime ti = 125 ps a free positron annihilation component, with intensity I2 and lifetime T2 and the o-Ps component, with intensity I3 and lifetime T3. Theory predicts the ratio /3//1 = 3, but as discussed in Chapter 11, certain effects may lead to a decrease in this ratio. The theoretical basis for relating the o-Ps lifetime to free volume is based on a model proposed by Tao [1972], in which < -Ps is assumed to be trapped in a potential well of... 

The free-volume concept dates back to the Clausius [1880] equation of state. The need for postulating the presence of occupied and free space in a material has been imposed by the fluid behavior. Only recently has positron annihilation lifetime spectroscopy (PALS see Chapters 10 to 12) provided direct evidence of free-volume presence. Chapter 6 traces the evolution of equations of state up to derivation of the configurational hole-cell theory [Simha and Somcynsky, 1969 Somcynsky and Simha, 1971], in which the lattice hole fraction, h, a measure of the free-volume content, is given explicitly. Extracted from the pressure-volume-temperature PVT) data, the dependence, h = h T, P), has been used successfully for the interpretation of a plethora of physical phenomena under thermodynamic equilibria as well as in nonequilibrium dynamic systems. 

Positron annihilation lifetime spectroscopy (PALS) is normally applied to determine the free volume properties of a cured thermoset. The theory and methodology of PALS [27, 28] is briefly described next. The positron, an antiparticle of an electron, is used to investigate the free volume between polymer chains. The birth of the positron can be detected by the release of a gamma ray of characteristic energy. This occurs approximately 3 ps after positron emission when the Na decays to Ne. Once inside the polymer material, the positron forms one of the two possible types of positroniums, an ort o-positronium or a p(3 ra-positronium, obtained by pairing with an electron abstracted from the polymer environment. The decay spectra are obtained by the death event of the positron, pi ra-positronium or ort o-positronium species. By appropriate curve fitting, the lifetimes of the various species and their intensity can be determined. The lifetime of an ort o-positronium (Xj) and intensity (I3) have been found to be indicative of the free volume in a polymer system because this is where the relevant species become localised. X3 is related to the size of the free volume sites and I3 to their number concentration. The free volume properties of difunctional and multifunctional epoxies are shown in Table 3.5. The data clearly... 

I peak-effects, physical and magnetic phase diagrams) -Raman, IR spectra -neutron diffraction -specific heat -thermal conductivity -positron annihilation (fundamental superconducting parameters, BCS theory)... 

Reaction with a polymerizable mixture, giving nanofibers covalently attached to the polymer, has also been studied [155]. Different techniques, including dynamic mechanical analysis and positron annihilation spectroscopy show that interaction at the nanofiber-polymer interface produces radical changes in the glass transition of the material. The effect of the addition of cellulose nanocrystals on the properties of a polyurethane matrix are theoretically described by the free volume theory. 

Apart from the theoretical approaches, electronic energy spectra of carbides and nitrides have been studied using a variety of experimental techniques X-ray emission and photoelectron spectrosopy, optical and Auger spectroscopy, electron energy loss and positron annihilation spectroscopy, etc. However, interpretation of the results obtained requires, as a rule, use of the computational methods of the band theory of solids and quantum chemistry. Moreover, the data provided by theoretical methods are important by themselves, because they give much more detailed information on the electron states and chemical bonding than any of the experimental methods. They also allow us to model theoretically... 

Therefore, the fraction of positrons annihilating in cavities with volumes between Vand V+dV is written as g(V)AV. Theoretical treatment using molecular dynamics and kinetic theory [146,147] has predicted that the radii and the free volume cavities in polymer obey the distribution functions J(J1) and g(V), respectively. 

Free-volume distributions of polystyrene probed by positron annihilation comparison with free volume theories. Macromolecules, 26, 7149-7155. 

In particular, the most powerful method for studing lattice defects, due to the high sensitivity of positrons to open volume defects such as vacancies, vacancy clusters, voids, dislocations, grain boundaries, etc., is positron annihilation spectroscopy (PAS). A diagram illustrating the applicability of PAS and other techniques as a function of defect size and density versus depth in material is shown in Figure 4.25. Thus, PAS represents a non-local experimental technique that is sensitive to microstructural defects at the atomic scale. A well-established theory of positron annihilation phenomena is currently available. Especially for metallic materials, it is possible to perform ab initio calculations of positron parameters for various defects and atomic arrangements [72,73]. 

Free volume present in nanocomposite systems plays a major role in determining the overall performance of the membranes. Positron annihilation lifetime spectroscopy (PALS) is an efficient technique used for the analysis of free volume. The diffusion of permeant through polymeric membranes can be described by two theories, namely, molecular and free-volume theories. According to the free-volume theory, the diffusion is not a thermally activated process as in the molecular model, but it is assumed to be the result of random redistributions of free-volume voids within a polymer matrix. Cohen and Turnbull developed the free-volume models that describe the diffusion process when a molecule moves into a void larger than a critical size, Vc- Voids are formed during the statistical redistribution of free volume within the polymer. It is found that the relative fractional free volume of unfilled polymer decreases on the addition of layered silicates. The decrease is attributed to the interaction between layered silicate and polymer because of the platelet structure and high aspect ratio of layered silicates. The decrease is explained to the restricted mobility of the chain segments in the presence of layered silicates. This results in reduced free-volume concentration or relative fractional free volume [49]. 

Meanwhile, the electron was found to have a positively charged counterpart called the positron the electron and positron could annihilate each other, with the emission of light quanta. The theory of the electron did in fact predict the existence of such a particle. It was later found that the existence of such opposite particles (antiparticles) was a much more general phenomenon than once surmised. 

In the quantum field theories that describe the physics of elementary particles, the vacuum becomes somewhat more complex than previously defined. Even in empty space, matter can appear spontaneously as a result of fluctuations of Ihe vacuum. It may be pointed, out, for example, that an electron and a positron, or antielectron, can be created out of the void, Particles created in this way have only a fleeting existence they are annihilated almost as soon as they appear, and their pressure can never be detected directly. They are called virtual particles in order to distinguish them from real particles. Thus, the traditional definition of vacuum (space with no real particles in it) holds. In their excellent paper, the aforementioned authors discuss how, near a superheavy atomic nucleus, empty space may become unstable, with the result that matter and antimatter can be created without any input of energy. The process may soon be observed experimentally. 

The positron was subsequently discovered by Anderson (1933) in a cloud chamber study of cosmic radiation, and this was soon confirmed by Blackett and Occhialini (1933), who also observed the phenomenon of pair production. There followed some activity devoted to understanding the various annihilation modes available to a positron in the presence of electrons radiationless, single-gamma-ray and the dominant two-gamma-ray processes were considered (see section 1.2). The theory of pair production was also developed at this time (see e.g. Heitler, 1954). 

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