Positron probability

Here ct, ce and cp are the concentrations of the positive ions, electrons and the positron probability density at a point r measured from the center of the blob at time t. Dp is the diffusion coefficient of the positron, Di = De = Damb 0 is the ambipolar diffusion coefficient of the blob, a2 o2 ss a2 is the dispersion of the intrablob species, and a2 is the dispersion of the positron space distribution by the end of its thermalization. Decay rate Te-1 = 1/t + kescs is the sum of the electron solvation rate and possible capture by solute molecules t 2 = 1 /t2 + l/r + kpscs accounts for the free e+ annihilation, solvation and reaction with S. Similarly, t 1 = l/rjmr + hscs, where T r is the rate of the ion-molecule reaction. 

Consider neon-18 or Ne-18. It has lOp and 8n, giving an n/p ratio of 0.8. For a light isotope, like this one, this value is low. A low value indicates that this isotope will probably be unstable. Neutron-poor isotopes, meaning that it has a low n/p ratio do not have enough neutrons (or has too many protons) to be stable. Decay modes that increase the number of neutrons and/or decrease the number of protons are favorable. Both positron emission and electron capture accomplish this by converting a proton into a neutron. As a rule, positron emission occurs with lighter isotopes and electron capture with heavier ones. 

Know that nuclear stability is best related to the neutron-to-proton ratio (n/p), which starts at about 1/1 for light isotopes and ends at about 1.5/1 for heavier isotopes with atomic numbers up to 83- All isotopes of atomic number greater than 84 are unstable and will commonly undergo alpha decay. Below atomic number 84, neutron-poor isotopes will probably undergo positron emission or electron capture, while neutron-rich isotopes will probably undergo beta emission. 

Each of the following nucleide classified as probably stable, beta emitter or positron emitter ... 

The nucleide near the belt of stability are probably stable, those above the belt are beta emitters, those below, positron emitters. Thus the stable hucleides are 208p 3 emitters are Ca, aI... 

Ii is possible for a positron-electron system to annihilate with the emission of one, two. three, or more gamma rays. However, not all processes are equally probable. 

As a final point in the introduction, it is interesting to note that the analogous process of positron capture by neutron excessive nuclei should be possible in principle. However, such captures are hindered by two important facts First, the number of positrons available for capture is vanishingly small in nature, and second, both the nucleus and the positron are positively charged and will repel one another. Compare this to the situation for electron capture in which the nucleus is surrounded by (negative) electrons that are attracted to the nucleus, of course, and the most probable position to find any s electrons is at the nucleus (r = 0). 

In his intervention Blackett treated the discovery of the positron in cosmic rays by C. D. Anderson in 193246 and its confirmation by Blackett and Occhialini,47 who had introduced, for the first time, the technique of triggering a vertical cloud chamber by means of the coincidence between two Geiger counters, one placed above, the other below the chamber. Blackett also discussed a number of papers by Meitner and Philipp, Curie-Joliot, Blackett, Chadwick and Occhialini, and Anderson and Nedder-meyer,48 all appearing almost at the same time, on the production of positrons in various elements irradiated with the -/-rays of 2.62 MeV energy of The. These were the first observations of electron-positron pair production. He also pointed out that the observed production of positrons has a cross section larger than the nuclear dimensions, and therefore, most probably, does not originate from a nuclear process. 

Consider spherical filler particles (phase f) in a matrix (phase m). The probability of a positron hitting a filler particle is proportional to the volume fraction of filler vr. The probability of the positron thermalizing and annihilating in this filler particle can be written [21] as... 

Inserting values for D+ and r into the expression for L+, a value of approximately 1000 A is obtained, and this is typical for metals. One can then find an estimate of the efficiency e of the moderator by multiplying the implantation profile P(x), equation (1.9), by the probability that a positron reaches the surface from a depth x, exp(—x/L+), and integrating over all values of x. Then, since pimpL+ -C 1, the efficiency may be written as... 

Perhaps of more general applicability for the study of the properties of positronium is its production by the desorption of surface-trapped positrons and by the interaction of positrons with powder samples. According to equation (1.15) it is energetically feasible for positrons which have diffused to, and become trapped at, the surface of a metal to be thermally desorbed as positronium. The probability that this will occur can be deduced (Lynn, 1980 Mills, 1979) from an Arrhenius plot of the positronium fraction versus the sample temperature, which can approach unity at sufficiently high temperatures. The fraction of thermally desorbed positronium has been found to vary as... 

Probably the most accurate positron-hydrogen s-wave phase shifts are those obtained by Bhatia et al. (4974), who avoided the possibility of Schwartz singularities by using a bounded variational method based on the optical potential formalism described previously. These authors chose their basis functions spanning the closed-channel Q-space, see equation (3.44), to be of essentially the same Hylleraas form as those used in the Kohn trial function, equation (3.42), and their most accurate results were obtained with 84 such terms. By extrapolating to infinite u in a somewhat similar way to that described in equation (3.54), they obtained phase shifts which are believed to be accurate to within 0.0002 rad. They also established that there are no Feshbach resonances below the positronium formation threshold. 

Several other calculations of the first few partial-wave phase shifts for positron-helium scattering have been carried out using a variety of approximation methods in all cases, however, rather simple uncorrelated helium wave functions have been used. Drachman (1966a, 1968) and McEachran et al. (1977) used the polarized-orbital method, whereas Ho and Fraser (1976) used a formulation based on the static approximation, with the addition of several short-range correlation terms, to determine the s-wave phase shifts only. The only other elaborate variational calculations of the s-wave phase shift were made by Houston and Drachman (1971), who employed the Harris method with a trial wave function similar to that used by Humberston (1973, 1974), see equation (3.77), and with the same helium model HI. Their results were slightly less positive than Humberston s HI values, and are therefore probably less... 

The most accurate theoretical results for positronium formation in positron-helium collisions in the energy range 20-150 eV are probably those of Campbell et al. (1998a), who used the coupled-state method with the lowest three positronium states and 24 helium states, each of which was represented by an uncorrelated frozen orbital wave function... 

A semi-quantitative picture of positronium formation in a spur in a dense gas was developed by Mogensen (1982) and Jacobsen (1984, 1986). If the separation of the positron from an electron is r, and there is assumed to be only one electron in the spur (a so-called single-pair spur), then the probability of positronium formation in the spur, in the absence of other competing processes, can be written as [1 — exp(—rc/r)] here rc is the critical, or Onsager, radius (Onsager, 1938), given for a medium of dielectric constant e by... 

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