Annihilating Electrons

Besides of creation operators, the second quantized formalism also requires to the use of formal operators which remove (annihilate) electrons. 

Consider first a one-electron system. An annihilation operator aj is defined so that it removes the electron from spinorbital Vj if the latter was occupied  

This last equation expresses that one cannot annihilate an electron from the vacuum state. 

That is, an electron is created on the vacuum in the state xi/j by aj , then it is annihilated by the operator aj and the result is again the vacuum state. 

One has to be careful when allowing an annihilation operator to act on a multi-electron state. In the case of a two-electron state, for example, one may write  

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The probability that the annihilating electron-positron pair has a momentum p is proportional to... 

The ability of angular correlation and Doppler broadening techniques to provide information concerning the momentum of an annihilating electron-positron pair was briefly discussed in section 1.3. Also, it... 

In terms of the creation-annihilation electron and phonon operators the Hamiltonian can be cast as follows ... 

A. Weiss, Positron Annihilation Electron Spectroscopy, in Handbook of Surface Imaging and Visualization, A.T. Hubbard, Ed., Chapter 45, pp. 617-632, CRC Press (1995). 

The Feynman diagram for the simplest annihilation event shows that annihilation is possible when the two particles are Ax h/mc 10 12 5 m apart, and that the duration of the event is At h/mc2 10-21 s. The distance is the geometric mean of nuclear and atomic dimensions, which is probably not significant. The distance is so much smaller than electronic wave functions that it may be assumed to be zero in computations of annihilation rates. The time is so short that, during it, a valence electron in a typical atom or molecule moves a distance of only ao/104, so that a spectator electron can be assumed to be stationary and the annihilating electron can be assumed to disappear in zero time. Thus the calculation of annihilation rates requires the evaluation of expectation values of the Dirac delta function, and the relaxation of the daughter system (post-annihilation remnant) can be understood with the aid of the sudden approximation [4], These are both relatively simple computations, providing an accurate wave function is available. 

The mean kinetic energies of annihilating electrons may provide information on the electronic surroundings where the positrons undergo annihilation. 

In general iV-electron configurations p) are obtained by annihilating electrons in M orbitals with Wp=l in 0) and creating electrons in M different orbitals with Wp=0 in 0). In other words we change the occupation numbers from 1 to 0 in M orbitals and from 0 to 1 in M different orbitals. 

We now introduce the very important concept of a one-hole state. Th is a state i) of the weak-coupling basis y) that is formed by annihilatin an electron in the target ground state 0). Clearly (y a(q) 0) in (11.13) is zero if ) is not a one-hole state. A one-hole state a) is denoted by the orbital of the annihilated electron. In the context of the many-electron ion space this notation cannot be confused with that of the orbital. 

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... 

The energy deviation is proportional to the momentum componentp//of the annihilating electron, which is parallel to the propagation direction of photons ... 

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) ... 

Lineshape spectroscopy is based on the Doppler shift of the energy of the annihilation radiation 72 (see fig. 1). The origin of this shift is the finite momentum p of the annihilated electron-positron pair in the direction of the emitted y-ray ... 

It is essential now to use this form of the second quantized Hamiltonian which is expressed over the true fermion operators obeying the anticommutation relations of Eqs. (13.13). The use of Eqs. (13.28) or (13.33) for the Hamiltonian would complicate the following treatment since the appearance of the overlap matrix in the commutation rules would destroy the purely algebraic character of the creation/annihilation operators. The proper anticommutation rules permit us to consider and as abstract operators creating and annihilating electrons, respectively. Accordingly, taking the variation of Eq. (14.8) we get ... 

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