Particle annihilation

Positrons cannot be observed directly because, as Figure 22-6a illustrates, when a positron encounters an electron, the two particles annihilate each other, converting their entire mass into a pair of photons. The occurrence of positron emission can be inferred from the observation of such a pair of photons. Each photon produced in this process has a specific energy Epi ton = 9.87 X lO kJ/mol. Photons with such high energy are called y rays. 

The analogous contractions that place the -particle annihilation operator to the right of the -particle creation operators are zero ... 

The form [Eq. (3)] of the perturbation operator points out that formally we obtain a double perturbation expansion with the two-electron V2 and one-electron Vi perturbations. However, in the case of a Hartree-Fock potential the one-electron part of the perturbation is exactly canceled by some terms of the two-electron part. This becomes more transparent when we switch to the normal product form of the second-quantized operators2-21 indicated by the symbol. ... We define normal orders for second-quantized operators by moving all a ( particle annihilation) and P ( hole annihilation) operators to the right by virtue of the usual anticommutation relations [a b]+ = 8fl, [i j] = 8y since a 0) = f o) = 0. Then... 

A proper definition of (quasi-)particle-creation and (quasi-)particle-annihilation operators an and a is provided by diagonalization of the (time-independent) unperturbed part Ho = //ext+Z/e-e of the total Hamiltonian. After the iteration is performed (e.g. on the Dirac-Fock level) the latter may be cast into the form... 

Fig. 3.4. (a) The electric field dose to the proton (composed of three quarks) is so strong that it creates matter and antimatter (shown as electron-positron pairs). The three quarks visible in scattering experiments represent the valence quarks, (b) One of the radiative effects in the QED correction of the c order (see Table 3.1). The pictures show the sequence of the events from left to the right A photon (wavy line on the left) polarizes the vacuum and an electron-positron pair (solid lines) is created, and the photon vanishes. Then the created particles annihilate each other and a photon is created, (c) A similar event (of the order in QED), but during the existence of the electron-positron pair the two particles interact by exchange of a photon, (d) An electron (horizontal solid line) emits a photon, which creates an electron-positron pair, that annihilates producing another photon. Meanwhile the first electron emits a photon, then first absorbs the photon from the annihilation, and afterwards the photon emitted by itself earlier. This effect is of the order c in QED. 

Positron emission occurs when an unstable nucleus emits a positron (Figure 19.4 ). A positron is the antiparticle of the electron it has the same mass as an electron, but the opposite charge. If a positron collides with an electron, the two particles annihilate each other, releasing energy in the form of gamma rays. In positron emission, a proton is converted into a neutron and emits a positron. 

Monte Carlo simulations require less computer time to execute each iteration than a molecular dynamics simulation on the same system. However, Monte Carlo simulations are more limited in that they cannot yield time-dependent information, such as diffusion coefficients or viscosity. As with molecular dynamics, constant NVT simulations are most common, but constant NPT simulations are possible using a coordinate scaling step. Calculations that are not constant N can be constructed by including probabilities for particle creation and annihilation. These calculations present technical difficulties due to having very low probabilities for creation and annihilation, thus requiring very large collections of molecules and long simulation times. 

The j3 -particles that are emitted in the j3 -decay mode are slowed down in the material around the source. When these reach very low velocities they interact with an ordinary electron and the pair is annihilated. The corresponding energy of 2 x E, or 1022 keV, is normally released in the form of two photons of 511 keV each, emitted in opposite directions. 

In addition to Compton scattering, y-rays having energies above 1022 keV interact with matter by a process called pair production, in which the photon is converted into a positron and an electron. The y-ray energy in excess of the 1022 keV needed to create the pair is shared between the two new particles as kinetic energy. Each j3 -particle is then slowed down and annihilated by an electron producing two 511-keV photons. 

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 study of the behavior of reactions involving a single species has attracted theoretical interest. In fact, the models are quite simple and often exhibit IPT. In contrast to standard reversible transitions, IPTs are also observed in one-dimensional systems. The study of models in ID is very attractive because, in some cases, one can obtain exact analytical results [100-104]. There are many single-component nonequilibrium stochastic lattice reaction processes of interacting particle systems [100,101]. The common feature of these stochastic models is that particles are created autocatalytically and annihilated spontaneously (eventually particle diffusion is also considered). Furthermore, since there is no spontaneous creation of particles, the zero-particle... 

We might as well attempt to introduce a new planet into the solar system, or to annihilate one already in existence, as to create or destroy a particle of hydrogen. All the changes we can produce consist in separating particles that are in a state of combination, and joining... 

The operator bx annihilates a particle from the A-state, while the operator 6J creates a particle in the A-state, leaving the other states unchanged the total population of the system changes by unity in each case. The numerical factors are chosen so that the product of the two operators in the appropriate order is given by... 

One speaks of Eqs. (9-144) and (9-145) as a representation of the operators a and o satisfying the commutation rules (9-128), (9-124), and (9-125). The states 1, - , ) = 0,1,2,- are the basis vectors spanning the Hilbert space in which the operators a and oj operate. The representation (9-144) and (9-145) is characterized by the fact that a no-particle state 0> exists which is annihilated by a, furthermore this representation is irreducible since in this representation a(a ) operating upon an n-particle state, results in an n — 1 ( + 1) particle state so that there are no invariant subspaces. Besides the above representation there exist other inequivalent irreducible representations of the commutation rules for which neither a no-particle state nor a number operator exists.8... 

A state of m particles and n antiparticles can be constructed from the no-particle state 0>, which now is annihilated by both the b and the On operators ... 

In formulating the second-quantized description of a system of noninteracting fermions, we shall, therefore, have to introduce distinct creation and annihilation operators for particle and antiparticle. Furthermore, since all the fermions that have been discovered thus far obey the Pauli Exclusion principle we shall have to make sure that the formalism describes a many particle system in terms of properly antisymmetrized amplitudes so that the particles obey Fermi-Dirac statistics. For definiteness, we shall in the present section consider only the negaton-positon system, and call the negaton the particle and the positon the antiparticle. 

This confirms our interpretation of the operators 6,6 and d,d as creation and annihilation operators for particles of definite momentum and energy. Similar consideration can be made for the angular momentum operator. The total electric charge operator is defined as... 

The representation of these commutation rules is fixed by the requirement that there exist no-particle states 0)in and 0>out which are annihilated by the corresponding positive frequency operators, i.e. 

Normal product of free-field creation and annihilation operators, 606 Normal product operator, 545 operating on Fermion operators, 545 N-particle probability distribution function, 42... 

Since Rutherford s work, scientists have identified other types of nuclear radiation. Some consist of rapidly moving particles, such as neutrons or protons. Others consist of rapidly moving antiparticles, particles with a mass equal to that of one of the subatomic particles but with an opposite charge. For example, the positron has the same mass as an electron but a positive charge it is denoted 3 or f e. When an antiparticle encounters its corresponding particle, both particles are annihilated and completely converted into energy. Table 17.1 summarizes the properties of particles commonly found in nuclear radiation. 

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