Open AccessThe annihilation of fast positrons by electrons in the K-shell
Author(s) -
H. J. Bhabha,
Harold Hulme
Publication year - 1934
Publication title -
proceedings of the royal society of london. series a, containing papers of a mathematical and physical character
Language(s) - English
Resource type - Journals
eISSN - 2053-9150
pISSN - 0950-1207
DOI - 10.1098/rspa.1934.0184
Subject(s) - physics , positron , electron , annihilation , atomic physics , cosmic ray , ionization , kinetic energy , atom (system on chip) , photoelectric effect , annihilation radiation , nuclear physics , quantum mechanics , ion , positron annihilation , computer science , embedded system
The probability of the simultaneous of a positron and an electron, with the emission of two quanta of radiation, has been calculated by Dirac and several other authors. From considerations of energy and momentum it follows that an electron and positron can only annihilate one another with the emission of one quantum of radiation in the presence of a third body. An electron bound in an atom could, therefore, annihilate a positron, represented by a hole on the Dirac theory, by jumping into a state of negative energy which happens to be free, the nucleus taking up the extra momentum. The process is now mathematically analogous to the photoelectric transitions to states of negative energy in the sense that the matrix elements concerned are the same, and we might expect that the effect would be most important for the electrons in the K-shell. Fermi and Uhlenbeck have calculated the process approximately, for the condition where the kinetic energy of the positron is of the order of magnitude of the ionization energy of the K-shell. The result they obtained was very small compared with the two quantum process, which is to be explained by the fact that for these small energies, the positron does not get near the nucleus. In view of the fact that positrons of energies of the order 100mc 2 occur in considerable quantities in the showers produced by cosmic radiation, and that the primary cosmic radiation itself may consist, in part, of positrons, it becomes of interest to calculate the cross-section for the annihilation of positrons of high energy by electrons in the K-shell, and their absorption in matter, and also to compare this process with the two quantum process for high energies. In the photoelectric effect for hardγ -rays, the electron the electron leaves the atom in states of different angular momentum (described by the azimuthal quantum numberl ), and the terms which give the largest contribution are roughly those for whichl is of the order of the energy of theγ -ray in terms ofmc 2 . For high energies, therefore, a calculation by the method of Hulme, in which the last step is carried out numerically, is out of the question, and we must find some approximate method of effecting a summation. We shall use an adaptation of Sauter's method, in which we shall treat as small the product of the fine structure constant and the nuclear charge. This method may be expected to give a good approximation for small nuclear charge. Our method has the further restriction that it is valid only when the kinetic energy of the positron isnot small compared withmc 2 .