Electron-positron momentum density

The sample is mounted on a cold finger—preferably cooled by a closed-cycle helium refrig-erator to 10 K—which greatly reduces the thermal smearing of the electron-positron momentum density. The sample is again tilted so that each detector is irrad-iated fairly uniformly, irrespective of gamma emission angle. 

Mijnarends, P.E., (1979) Electron momentum densities in metals and alloys. In Positrons in Solids, (Ed.) Hautojarvi, P., Springer. 

Several experimental techniques such as Compton scattering, positron annihilation, angular correlation, etc., are used for measuring momentum densities. One of the most popular techniques involved in measuring momentum densities is termed as electron momentum spectroscopy (EMS) [29]. This involves directing an electron beam at the surface of the metal under study. Hence EMS techniques fall under what is classified as coincidence spectroscopy. 

As most experimenters want to know not the absolute value of the linewidth but how it changes as a function of physical parameters, these ratios have been taken up as the simplest way of describing the linewidth. C/T is called the S (sharpness) parameter, and (A+E)/T is W, the wing parameter. As we can induce from Figure 3.7, S and W should be sensitive to changes in the momentum density of lower- and higher-momentum electrons, respectively. Positron annihilation in open volume defects thus typically leads to an increase in 5 and a decrease in W. 

From the invariance of the action S with respect to translations in 4-coordinate space follow the expressions for the energy and momentum densities of the electron-positron field ... 

Positron annihilation techniques [78] can be used to obtain information about the momentum density of the annihilating positron-electron pair. In solids, particularly metals, the distortion of the electron momentum density by the Coulomb interaction between the positron and electrons is relatively small, and this technique then gives us the electron momentum density. 

Barnes and Peter (1989) presented a formal theory of positron aimihilation in superconductors. They showed that, in a superconducting state arising through pair formation in a non-interacting electron gas, the two-photon momentum density Psc(.P), in the absence of electron-positron correlations, is described by... 

As the positron is thermalized, px and py are, to a good approximation, momentum components of the annihilated electron. The 2D-ACAR measurement yields N(px,Py), the projection of the two-photon momentum density p (p) ... 

As the study of the electron momentum density and the Fermi surface by the positron annihilation techniques is rather indirect, it is important to perform the experimental investigations in close relation with a strong computational approach. The first valuable quantity to estimate is the positron density distribution in the lattice unit cell it will indicate which electronic states might be observed experimentally. One has then to evaluate the electron momentum density as seen by the positron. From this, the... 

After investigating the positron behavior, the next step is to solve the band structure and to calculate the momentum density p (p). It is rather controversial to describe the electronic structure of high-Tc superconductors by band structure. But it is almost the only way to obtain a calculated momentum density in order to interpret the positron measurements. We refer to the chapter by Pickett and Mazin in this volume (ch. 193) for a general discussion of band-structure calculations in superconducting cuprates. Studies in relation with positron annihilation have been made using different methods FLAPW (Massidda 1990, Massidda et al. 1991), LMTO (Bansil et al. 1988, Barbiellini et al. 1992,... 

Hence, while the lifetime method yields information regarding the electron density in the region of electron-positron overlap through the determination of positron decay rates from various states in the metal, the method of angular correlation and Doppler broadening (the momentum technique) gives information regarding the... 

When positrons enter a material they annihilate with electrons in the material, giving rise coincidentally to two annihilation y-rays of -511 keV (equal to the electron or positron mass) at nearly 180° apart (Fig. 9.21). The annihilation occurs in a time (lifetime) after their emission from the positron source which depends on the density of electrons in the metal around the positron at the time of annihilation. The energy spread of the y-rays and the angle between them depend on the momentum of the annihilated electron. There are several experimental methods that can be used to measure these quantities including positron lifetime, positron annihilation... 

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]. 

The shape of the angular correlation distribution profile is a superposition of two contributions. Annihilation of the positron with the conduction electrons results in a contribution that has the approximate shape of an inverted parabola, while the contribution of the high-momentum core electrons is reflected in a broad, Gaussian-like distribution. When the core electron density is reduced, as is the case for a positron trapped in a vacancy, the contribution of the conduction electrons is enhanced and a narrower distribution is measured. Therefore, changes in the shape of the angular correlation distribution profile reflect a reduction in annihilation with core electrons [127]. 

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