Electron annihilation

A positron has the same mass as that of an electron, but the opposite charge. When a positron emitted in a PET scan encounters an electron, annihilation occurs in the body electromagnetic energy is produced and no matter remains. How much energy (in joules) is produced in the encounter See Box 17.1. 

C22-0091. A positron has the same mass as an electron. When a positron and an electron annihilate, both masses are converted entirely into the energy of a pair of y rays. Calculate the energy per y ray and the energy of I mol of y rays. 

It was found that over a narrow free energy range rapidly increases with asymptotically approaching a limiting value smaller than . Although the observed behavior remains in accordance with the Marcus model prediction, it was, somewhat unexpectedly, found that not the electron annihilation energetics is a only factor... 

The action of operator a reduces nj from 1 to 0 if spin orbital i is occupied and gives zero if spin orbital i is unoccupied. The operators aj are for that reason called electron annihilation operators. A special case of Eq. (1.15) is... 

It should be stressed that in the literature one can come across a wide variety of notations for creation and annihilation operators. In this book we follow the authors [14, 95] who attach the sign of Hermitian conjugation to the electron annihilation operator, but not to the electron creation operator. Although the opposite notation is currently in wide use, it is inconvenient in the theory of the atom, since it is at variance with the common definitions of irreducible tensorial quantities. 

Electron annihilation operators, as Hermitian conjugates of creation operators, are no longer the components of an irreducible tensor. According to (13.40), such a tensor is formed by (21 + l)(2s + 1) components of the operator... 

We shall now introduce the electron creation operators a and b and the electron annihilation operators a = (—b = (—1 y mb for the electrons in the subshells nihjNl and n2hjNl, respectively. They are irreducible tensors of rank t = 1 /2... 

A iV-electron vacancy (hole) in a shell may be denoted as nl N = ni4l+2-N (see aiso Chapters 9, 13 and 16). As we have seen in the second-quantization representation, symmetry between electrons and vacancies has deep meaning. Indeed, the electron annihilation operator at the same time is the vacancy creation operator and vice versa instead of particle representation hole (quasiparticle) representation may be used, etc. It is interesting to notice that the shift of energy of an electron A due to creation of a vacancy B l is approximately (usually with the accuracy of a few per cent) equal to the shift of the energy of an electron B due to creation of a vacancy A l, i.e. 

The total positron scattering cross section, erT, is the sum of the partial cross sections for all the scattering channels available to the projectile, which may include elastic scattering, positronium formation, excitation, ionization and positron-electron annihilation. Elastic scattering and annihilation are always possible, but the cross section for the latter process is typically 10-2O-10-22 cm2, so that its contribution to erT is negligible except in the limit of zero positron energy. All these processes are discussed in greater detail in Chapters 3-6. 

If sufficient positrons can be confined, studies of particle transport within the plasma, etc., similar to those conducted with electrons can be carried out. It may be possible to use the enhanced detection possibilities afforded since positron-electron annihilations can be detected. An ultra-cold source of positrons would also have a variety of other applications.24 For example, it has been proposed to eject trapped positrons into a plasma as a diagnostic.25 Also, positrons initially in thermal equilibrium at 4.2K within a trap would form a pulsed positron beam of high brightness when accelerated out of the trap. 

The26A1 decay in the interstellar medium also emits positrons (antielectrons), which may be a partial source of the positron-electron annihilation into gamma rays that are also seen emanating from the interstellar gas. 

Positron emission tomography (PET) exploits the difference in positron-electron annihilation rates in the reaction ... 

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. 

A positron and an electron annihilate each other upon colliding, thereby producing energy ... 

Note that p exits an F vertex (i.e., p represents electron creation or hole annihilation), and q enters an F vertex (i.e., q represents electron annihilation or hole creation). Also, p and q exit V vertices, and r and s enter but p and q enter T vertices, while r and s exit. Since V and T vertices are degenerate, the assignment of values to these vertices is determined only up to phase [cf. Eqs. (85)—(87)]. It will expedite the establishment of the overall phase factor of the matrix element (cf. Rule 3 later) if antisymmetrized component matrix elements with the same ordering of indices as the corresponding direct matrix elements are used. Hence, p and r, q and s, etc., are associated or considered participants in the same basic interaction. 

It is convenient to introduce an electron annihilation operator, written as a, which destroys an electron in spin orbital to give an (N — l)-electron ket if spin orbital , is occupied, and which gives zero if the orbital is unoccupied in the ket. For example... 

It may be readily verified that the matrix exp( - iA) is in fact unitary, provided the matrix A is Hermitian. The fact that the matrix exp( — iA) is unitary also means that the operator exp( — iA) is unitary and its matrix representation in the full ket expansion space, with matrix elements , is a unitary matrix. An analogous relation holds for transformations of the electron annihilation operators a, but it is the creation operator expansion that is most important for the MCSCF method. Substitution of the operator transformation into the expression of an arbitrary determinant gives the relation... 

Figure 4.22 The working principle of positron emission tomography (PET) is based on the decay of F-labeled diagnostics, for example [ F]2-fluorodeoxyglucose ([ F]FDG) [85]. The y photon pairs resulting from positron-electron annihilation are detected left) and enable spatial resolution of the sites where the labeled diagnostics and their congeners are predominantly processed. [ F]FDG is particularly useful for identification of metabolically active areas with high glucose turnover, for example brain tumors (nght). The two PET scans show a healthy brain above) and a newly developed tumor below, arrow courtesy of Hamamatsu Photonics).

The subscript i labels the principle quantum number and angular momentum quantum numbers (njlm). Here a,- and bj denote electron-annihilation and positron-creation operators, respectively, defined via diagonaiization of the unperturbed Hamiltonian (1.11)... 

Here, i CT(r) and (r) represent the ordinary nonrelativistic electron annihilation and creation operators. An LDA-type approximation has recently been derived for the exchange-correlation free energy Fxc[n, xl leading to explicit expressions for the effective potentials Veg(r) and Aeff (r, r ) (Kurth et al. 1999). 

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