Spontaneous Emission Probabilities at Radio Frequencies

Abstract

For nuclear magnetic moment transitions at radio frequencies the probability of spontaneous emission, computed from A_u = (8\pi u^3)hu(8\pi^3u^2/3h^2) sec^{-1}, is so small that this process is not effective in bringing a spin system into thermal equilibrium with its surroundings. At 300K, for u=10^7 sec.^{-1}, \mu=1 nuclear magneton, the corresponding relaxation time would be SX10^{21} seconds! However, for a system coupled to a resonant electrical circuit, the factor 8\piu^2/c^3 no longer gives correctly the number of radiation oscillators per unit volume, in unit frequency range, there being now one oscillator in the frequency range u/Q associated with the circuit. The spontaneous emission probability is thereby increased, and the relaxation time reduced, by a factor f=3Q\lambda^3/4\pi^2V, where V is the volume of the resonator. If a is a dimension characteristic of the circuit so that V~a^3, and if \delta is the skin-depth at frequency u, f~\lambda^3/a^2\delta. For a non-resonant circuit f~\lambda^3/a^2, and for a &8 it can be shown that f 'A'/ab'. If small metallic particles, of diameter 10 ' cm are mixed with a nuclear-magnetic medium at room temperature, spontaneous emission should establish thermal equilibrium in a time of the order oF minutes, for v=107 sec.