Thermalised positrons

Figure 3.4 illustrates two lifetime spectra collected by methods similar to those outlined above, (a) exhibits the non-exponential shoulder region associated with the annihilation of non-thermalised positrons. After thermalisation (essentially at time zero for condensed matter) the spectra are sums of exponential components associated with each decay mode, and a background component B, A] = 2 A, exp(-Ajt,) + B. For long lifetime components (> Ins) each X can be extracted by non-linear least squares fitting. For short X values characteristic of condensed matter, however, a... 

Moderation efficiency could be greatly enhanced by drifting a larger fraction of thermalised positrons to the exit surface. Attempts to realise field-assisted moderation have to date largely foundered because of the interactions of positrons with the interfaces between the material across which an electric field is maintained and the conductive coatings to which the potentials are applied. One reported observation of the enhancement of positron emission by an electric field has been that from a solid gas moderator whose surface was charged by electron bombardment [44]. 

The predominant annihilation process for thermalised positrons is via the direct production of two photons (Fig. 1). If both the positron and... 

Diffusion of thermalised positron and subsequent trapping at an open volume defect ( 10 s)... 

When a positron is emitted from a source and then penetrates into a solid, it quickly loses kinetic energy until it reaches the thermal level (Figure 4.26). This thermalised positron moves around in the solid by diffusion and finally annihilates with an electron. 

When a positron is emitted from a source, and penetrates into a solid, it quickly loses its kinetic energy to thermal energy. The thermalised positron moves around in the solid by diffusion and finally annihilates with one of the electrons in its surroundings. All of the energy from the electron-positron annihilation is converted into two annihilation y-rays, which can be detected. The annihilation rate of a positron is determined by the local electron density in the locale of the positron. Thus, positrons... 

One can immediately conclude that thermalisation times are very short compared to positron bulk lifetimes tb. The fact that positrons reach thermal energies very quickly is important for the application of the positron method. Clearly only thermalised positrons annihilate. The momentum of positrons is thus very small compared to the momentum of the electrons with which they annihilate. It is evident from Table 4.14 that positron scattering off phonons occupies more than 50% of the thermalisation time. Therefore, the positron spends most of the slowing-down process with an energy slightly above the thermal energy. This fact is important for the possible trapping of non-thermalised positrons. 

The thermalised positron is further scattered by phonons and diffuses until it annihilates with an electron (or is eventually trapped). During its lifetime (r > 10 s), it can diffuse over a volume of about 1000 A [83]. The diffusion process of a positron e" " can by characterised by a diftiisivity D+ and a mobility /x+, which are related by the Einstein equation... 

Furthermore, we assume that there is no trapping of non-thermalised positrons i.e.,... 

When a positron has reached thermal equilibrium with the medium, its scattering is overwhelmingly dominated by phonons. This scattering is usually quasi-elastic and does not affect the average positron momentum distribution. Thus, the momentum distribution of a thermalised positron has a time-independent form which is close to the Maxwell-Boltzmann distribution [72]. The average value of the square of the positron s thermal velocity is... 

In the classical picture, the movement of the thermalised positron is a near-isotropic random walk [91]. The mean free path of thermalised positrons (i.e., the mean distance between two scattering events) is... 

This means that a positron travels the furthest (on average hundreds of pm) in a metal during its thermalisation, which in turn ensures that PAS provides non-local information on the microstructure of the studied material. A thermalised positron in a defect-free material travels a distance of hundreds of nm, its mean free path is a few nm, and its wavelength is a few tenths of a nm. 

Another relatively recent technique, in its own way as strange as Mossbauer spectrometry, is positron annihilation spectrometry. Positrons are positive electrons (antimatter), spectacularly predicted by the theoretical physicist Dirac in the 1920s and discovered in cloud chambers some years later. Some currently available radioisotopes emit positrons, so these particles arc now routine tools. High-energy positrons are injected into a crystal and very quickly become thermalised by... 

The measurement of the mean lifetimes of positrons in matter has been one of the cornerstones of positron science over the past half-century. The lifetime of a positron in matter—gas, liquid or solid—will depend on the electronic environment in which it finds itself, and this in turn tells us much about the submicroscopic nature of the material. In condensed matter a positron will approach thermal energies within about lps, so that measured lifetimes are essentially those of a thermal positron in the material under study. In some gaseous environments—particularly in the noble gases—the time taken for a positron to come to thermal equilibrium with its surroundings is much longer—10°-102 ns—and this thermalisation time has to be taken into account in the analysis of time spectra. 

Simultaneous measurement of positron lifetime and the momentum of the annihilating pair can give information on thermalisation and transitions between positron states (and hence on chemical reactions of positrons or Ps). The most recent version uses MeV positron beams [35]. A full description of AMOC can be found elsewhere in this volume. 

Positron beams essentially separate the thermalisation of positrons implanted into a material from their eventual annihilation in another. While the field has been enlivened by a number of ingenious and exciting experiments— LEPD, PAES, etc. (reviews are to be found in 4, 35, 41, 42), in this section we shall concentrate on the basic elements of positron beam experimentation. 

Perkins and Woll (1969) have investigated the possibility of observing the effects of superconductivity on positron thermalisation. If the positron lifetime in the superconducting compound is small, they conclude that a lack of thermalisation of the positron may be observed. This would be reflected, for example, in a loss of resolution in an ACAR experiment. But the effect is small and has never been reported in low-Tc superconductors. It is doubtful that it might be observed in high-To superconductors because positron lifetimes are rather long in cuprates. 

Fig. 1. Schematic representation of positron annihilation spectroscopy. The positron e is emitted fiom the source simultaneously with the Y-ray y. When it annihilates after thermalisation with an electron of momentum p, the two y-rays Y, and Yz are emitted.

The mean diffusion length L+ of the positron is defined as the mean distance from the point of thermalisation that the positron reaches by diffusion movement. This quantity is related to the diffusion coefficient by the relation [72]... 

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