Annihilation photons

An important characteristic of positronic systems relevant to the experiment is their lifetimes. The expectation value of the electron-positron contact density allows us to evaluate the two-photon annihilation rate for a positronic system using the expression... 

In the single-mode Hamiltonian H, the quantities (fit) are the photon annihilation (creation) operators, respectively to is the frequency of the... 

Alekseev, A.I. (1958). Two-photon annihilation of positronium in the P-state. Sov. Phys. JETP 34 826-830. 

Farazdel, A. and Cade, P.E. (1977). The electronic structure and positron annihilation characteristics of positronium halides [X e+]. II. Two-photon annihilation. J. Chem. Phys. 66 2612-2620. 

Ore, A. and Powell, J.L. (1949). Three-photon annihilation of an electron-positron pair. Phys. Rev. 75 1696-1699. 

Equation (29) is particularly useful for the quantum theory because we know how to represent the Coulomb gauge vector potential operator in terms of photon annihilation and creation operators since the Green s function g(x,x ) remains a c-numbcr. (29) gives an operator representation of the vector potential in an arbitrary gauge. 

A.H. Hoang, P. Labelle, S.M. Zebarjad, Single photon annihilation contributions to the... 

The best theoretical value [17, 18] of T is 2.0861 ns-1. This two photon, Hylleras-type calculation also includes order-a corrections for 3-photon annihilation and 2-photon radiative corrections [17]. 

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 p-Ps has a shorter lifetime than o-Ps and it annihilates into two photons, while o-Ps annihilates into three photons. The intrinsic lifetime is 0.125 ns and 142 ns for the free p-Ps and o-Ps, respectively. In ordinary molecular media, the electron density is low enough so that Ps can pick off electrons from the media that have anti-parallel spin to that of the positron, and undergo two-photon annihilation. This is called the pick-off annihilation of Ps. The pick-off annihilation of o-Ps not only occurs in the form of two-photon annihilation, it also shortens the o-Ps lifetime from 142 ns (free o-Ps) to a few ns. The pick-off annihilation lifetime of o-Ps in molecular systems is about one order of magnitude greater than in crystalline or metallic media. Experimental determination of o-Ps lifetime is one of the most useful methods for positron and positronium chemistry. This is because o-Ps lifetime contains information about electron density, which governs the basic properties of chemical bonding in molecules. It is also controlled by the physical structure of molecules. 

A given system in a given state has only one annihilation rate. The terms two-photon annihilation rate, three-photon annihilation rate , and spin-averaged annihilation rate, sometimes seen in the literature, have no operational meaning and are not measurable [5]. 

The 3-to-2 photon ratio technique observes the ratio of 3 versus 2 photon annihilations of positronium (and positrons). In vacuum positronium will annihilate via 3-photon decays only. Trapped inside closed pores, both annihilation paths are possible. This change can be used to detect the onset of open porosity as shown in Figure 7.2. A change in slope (the slope is shown as a solid line) occurs at about 23% porogen load, indicative of an increased likelihood for positronium to escape from the sample. 

Figure 7.2 Ratio of 3 to 2 photon annihilations (O, left scale) when positrons are implanted at a mean depth of 100 nm as a function of porogen load. At a porogen load of 23% the slope increases, indicating positronium escapes from the samples. Beyond 50% positronium can also encounter the Si interface. The first derivative is shown as a solid line (right scale).

The ratio of 3-to-2 photon annihilations is not only a measure for open porosity it also changes with the concentration of pores and their size. A typical ratio measurement can be performed in several seconds on standard... 

Roughly speaking, 3-to-2 photon annihilation ratio measurements can be considered as a BET technique, which is sensitive to both open and closed pores. Positronium can be considered as the smallest atom possible. No pore will be too small. Positrons are implanted rather than adsorbed and forms positronium. Positronium annihilates into 2 or 3 photons from within pores, or into 3 photons only after escape (desorption) out of the sample through open porosity. In addition, depth dependent information can be provided. 

Positronium can pick-off an electron during a collision with a pore wall and annihilate into two photons. Between collisions, only three photon annihilations occur, just as in vacuum. Quantum mechanically, the overlap with the wall-electron wave functions decreases with the distance from the wall and pick-off (two photons) becomes less likely. With increasing pore size collisions become less frequent. The ratio of 3 photon annihilations to 2 photons probes the combination of pore size and total pore volume as well as their link to the sample surface, and can be measured by examining the energy distribution of annihilation photons. This 3-to-2 photon ratio can be calibrated to absolute fractions of positronium in the annihilation spectrum [16, 17]. 

Figure 7.5 3-to-2 photon ratio when considering the diffusion length, the formation probability and the increased chance for three-photon annihilation in larger pores. 

The fit results from 3-to-2 photon annihilation data can also be used to determine the fraction of the total porosity, which is open to the surface. The fraction is determined by comparing the asymptotic values of the fit for the total signal and the constant part. Here, too the implantation profile and the motion of positrons and positronium have to be taken into account. The results are shown in Figure 7.9. 

Para positronium two photon annihilations cause the narrow sharp feature in the center at zero momentum. The events at large momentum (> 4a.u.) are due to three photon annihilations. By adding lead filters between the sample and the detectors, the third photon is absorbed and only two photon events meet the sum energy restriction. 

Let us stress that the operational definition of the quantum phase of radiation [47] is also based on the use of bilinear forms in the photon operators. In the simplest form, the idea of the operational approach to the phase difference can be illustrated with the aid of the two-port interferometer shown in Fig. 11 (see Refs. 14 and 47 for more detailed discussion). The two incident monochromatic (or quasimonochromatic) light beams are combined by a symmetric beamsplitter oriented at 45° to each beam. The resultant intensities emerging from each output port are measured by the two photodetectors connected with a comparator (computer) as in the Hanbury-Brown-Twiss interferometer [85] (also see Refs. 14, 15, and 86). Following Noh et al. [47], we denote by a and 2 the photon annihilation operators, describing the field at the two input ports, and by a and 04 the corresponding operators at the two output ports. Then... 

The photon annihilation operators associated with the field at the output arms of PBSi are related to those at the input through... 

Here V denotes the quantization volume, and e 1 is the unit polarization vector for the radiation mode characterized by wavevector k, polarization A and circular frequency co = c k where it appears, an overbar denotes complex conjugation. The polarization vector is considered a complex quantity specifically to admit the possibility of circular or elliptical polarizations. Associated with each mode (k, A) are a Hermitian conjugate pair of photon annihilation and creation operators, and k / , respectively, which operate eigenstates of //raci with m(k, A) photons (m being the mode occupation number) as follows... 

Given that the molecule is initially in its ground state, there are initially q photons of the pump mode (k, X) and q photons of the harmonic mode (k, X ). There are three possible sequences of photon annihilation and creation (a, b, and c) that can provide a route from the initial to the final state, each involving different virtual intermediate states. To avoid confusion, the intermediate state labels r[Pg.619]

This constant is essentially the limit of K as gi en by Eq. (5.14), in the case where the two absorbed photons become identical however, the factor rtiniY is replaced by n (n — 1) since the photon annihilation operator acts twice on the same radiation mode. As will be seen below, this difference is ultimately reflected in a dependence on the coherence properties of the laser source, which is uniquely associated with single-beam processes. It is also worth observing that although the first two terms of Eq. (5.13) become identical if the two absorbed photons are deiived from the same beam, inclusion of a factor of 2 in Eq. (6.1) would amount to double-counting the time-ordered diagrams, and is therefore not ap])ropriate. 

It consists of a lattice of N molecular sites spaced by distance a and comprises the exciton part (an is the exciton annihilation operator on the site n), photon part (bk is the photon annihilation operator with the wavevector k and a given polarization) as well as the ordinary exciton-photon interaction. The cavity photon energy ek is defined by eqn (10.44), e represents the average exciton energy, while en are the on-site exciton energy fluctuations. 

The principle of the resonance experiment rests on the unequal populations of the M=0 triplet and the M= l triplet states in a magnetic field due to the admixture of singlet state in the M=0 triplet state in a magnetic field H, and on the increase in two photon annihilation which results when transitions are induced from the M= l to the M=0 triplet states. 

This can be accomplished by setting up appropriate kinetic equations arid subsequent integration of the resulting differential equations, from which the population of the various states in which the positrons exist o-Ps and PsM can be found as a function of time. From these values and the positron annihilation constants for these states, an equation for the time dependent two photon annihilation rate can be obtained, which in turn allows the determination of the chemical reaction rate constants by utilizing sophisticated nuclear chemical lifetime measurement techniques. 

In order to obtain rate constants for the reaction of Ps (or positrons) with substrate molecule or to follow changes in the reactivity of a certain medium towards Ps the two photon annihilation rate (see above) has to be determined. This is accomplished by positron lifetime conventional fast-slow y y coincidence methods. 22... 

The momentum conservation law rules the direction of photons, as well. In the case of two-photon annihilation, photons must depart collinearly and the three-photon annihilation determines a plane in which photons should stay. This strict rule is usually altered slightly by... 

In the case of two-photon annihilation, the energy conservation law has one more serious consequence on photons. Both should have an energy of 0.511 MeV. This energy is so uniquely characteristic to electron-positron annihilation that Anderson (1932) proved the existence of positrons by the detection of this annihilation radiation. Although the above rule is rather strict, it is altered, again, a little by the momentum of the electrons. 

Just as for sum-frequency mixing, the generated intensity at low conversion efficiency grows as the product of the pump intensities at co and 0)2- In this situation a photon is created at both 0)3 and 0)2 for every photon annihilated at cl>. The wave-vector mismatch is AA = kieoy) — [k ct) ) — k co2), and the phase-matching condition is... 

Now the field part of the steady state Hamiltonian is written in terms of photon creation (a+) and photon annihilation (a) operators... 

Fig. 8.5. Feynman diagrams for Bhabha scattering (e+e" — e+e"), muon production (e+e" — and two-photon annihilation.

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