Spontaneous Emission

the emission of electromagnetic radiation by atoms and other quantum systems in excited energy levels. In contrast to induced radiation, spontaneous emission does not depend on the effect of external electromagnetic radiation on the quantum system; the laws governing the emission depend entirely on the system’s properties, as is the case with such other types of spontaneous transformations as radioactive decay and the transformations of molecules in monomolecular reactions.

Spontaneous emission is observed during the spontaneous quantum jump of an excited system from a higher energy level E_i to a lower level E_k. It is characterized by the frequency v_ik of the emitted photon, with energy hv_ik = E_i – E_k (where h is Planck’s constant), and by the probability A_ik, equal to the average number of such photons emitted by the quantum system per unit time. If the number of atoms or molecules in an excited energy level E_i (the population of the level) is N_i, then the power of spontaneous emission—the energy of the photons emitted per second—is N_i A_ikhv_ik; the power determines the intensity of spontaneous emission, which remains constant for constant N_i. If the initial number of excited systems is given as N_i0 and there is no further excitation, then as a result of spontaneous emission a decrease in N_i will occur with time f according to the law N_i = N_i0exp (– A_it, where A_i is the total probability of spontaneous emission upon transitions of the system from energy level Ei to increasingly lower energy levels E_k(A_i = ƩA_ik). The larger the A_i the faster the spontaneous emission attenuates with time and the shorter the lifetime Ƭ = 1/A_i, in level E_i.

The probability A_ik of spontaneous emission, which is the most important characteristic of the quantum transition between energy levels Ei and E_k, depends on the properties of both levels. For dipole radiation, A_ik is proportional to the cube of the transition frequency and the square of the dipole transition moment; in the visible region of the spectrum, A_ik is ~10⁸ sec^–1, corresponding to a lifetime of ~10^–8 sec for excited energy levels. In spectroscopy the dimensionless probabilities f_ik = A_ik/A_a are often used in place of the probabilities A_ik. They are also referred to as oscillator strengths, and here A₀ is the probability, taken as uni ty, that gives the same attenuation law for spontaneous emission as for the dipole radiation of an elastically bound electron accord ing to classical theory.