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The quantum theory of radiation damping‡

Published online by Cambridge University Press:  24 October 2008

A. H. Wilson
A. H. Wilson
Affiliation:
Trinity CollegeCambridge
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Extract

A general method is given of calculating the effect of damping on the collision cross-sections for problems involving free electrons and mesons. The result is expressed in the form of an integral equation which can only be solved if certain simplifying assumptions are made. It is shown that radiation damping has a negligible effect on the scattering of light by free electrons, so that the Klein-Nishina formula is unchanged by the inclusion of damping effects. The effect of damping for mesons, on the other hand, is extremely large and reduces the cross-sections considerably. The main problems considered are the nuclear scattering of mesons and the energy radiated by mesons during collisions. The revised cross-sections are much more reasonable than those calculated previously, but on account of the inadequacy of the data no detailed comparison with the experimental results is possible.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 37 , Issue 3 , July 1941 , pp. 301 - 316
Copyright
Copyright © Cambridge Philosophical Society 1941

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References

§ Z. Phys. 63 (1930), 54.

∥ Ann. Phys., Leipzig, 9 (1931), 23.

‡ Proc. Roy. Soc. A, 172 (1939), 384; (1941), in the press.

§ Z. Phys. 88 (1934), 436.

∥ Acta Phys. Polonica, 7 (1838–1939), 159 and 374.

‡ It may be noted that (7) cannot be considered as satisfactory for large values of t, for as t tends to infinity a n tends to a non-zero limit, and this would denote a permanent excitation of the electron to the intermediate state. To obtain a solution valid for large t as well as for small, we should have to make the time factor in a n to be of the form

This would introduce an extra term into a f of the form

but this term gives a negligible contribution to the transition probability on account of the largeness of the denominator. To introduce such terms here would make the theory very complicated, and it is doubtful whether their introduction could be justified while other terms of comparable importance are neglected. (These terms are necessary when dealing with the resonance fluorescence, since there E i, E n and E f are all nearly equal.) The expression (7) can therefore only be considered valid for not too large values of t.

‡ Each intermediate state with given momentum really consists of four. In order not to complicate the notation unduly, we omit here the summations with respect to n 1, n 2 and n 3.

‡ The angular dependence is given by

and the scattering can be considered to be isotropic without introducing too much error. In the frame of reference in which the electron is initially at rest, the scattering is almost entirely small angle scattering.

‡ See, for example, Wilson, A. H., Proc. Cambridge Phil. Soc. 36 (1940), 363,CrossRefGoogle Scholar equations (29) and (30).

‡ See for example, A. H. Wilson, loc. cit. equations (41) and (47). For the collision of a positive meson and a neutron only the intermediate states n 1 are possible. For the collision of a negative meson and a neutron (or of a positive meson and a proton) a meson must be first emitted by the nucleon, and only the intermediate states n 2 and n 3 are possible. There is little difference between the cross-sections for these two cases.

‡ Proc. Roy. Soc. A, 175 (1940), 483.

‡ Booth and Wilson, loc. cit., equation (132).