Effects of Configuration Interaction on Intensities and Phase Shifts

One of the enduring goals of scientific work at the National Institute of Standards and Technology (NIST) has been the expression of measurements in terms related directly to natural phenomena of an invariant and absolute character. For example, the unit of time, the second, is now defined as exactly 9,192, 631, 770 periods of oscillation of the radiation associated with a specified quantum transition between states of the Cs atom. This makes it possible, in principle, for any laboratory to realize the value of the second by preparing a sample of Cs in conditions that make it resemble a group of identical atoms unperturbed by their immediate environment. Of course, such an ideal realization is not attainable in practice. This permanent fact of life has provided steady stream of work over the years for theoretical physicists at NIST: there is always a need for models that can provide better quantitative links between realistic and ideal situations. Ugo Fano (1912– ) is believed to be the first theoretical physicist hired by NIST, and he has certainly been one of the most influential to date. His 1961 paper Effects of Configuration Interaction on Intensities and Phase Shifts [1] is one of the most frequently referenced journal articles by a NIST author, having been cited over 3200 times in the scientific literature. This paper treats a subject of fundamental interest to metrology and physics: the excitation spectra of quantum-mechanical systems. Its key result, the simple formula given in Eq. (3) below, is now well known to physicists as the “Fano profile” or “Fano line shape.” It addresses the challenge of expressing observed phenomena in a concise manner that can be derived from first principles. The celebrity enjoyed by this formula derives from the basic importance of the systems it describes, its wideranging practical utility, and the historical context in which it emerged. These aspects are discussed in turn below, though they cannot be entirely disentangled. Among the phenomena key to the early development of quantum mechanics were atomic spectra, i.e., the colors of light absorbed or emitted by free atoms. Such light was found to consist largely of discrete frequencies whose distribution is a characteristic property of the atomic species involved (see Fig. 1). The existence of these discrete frequencies led Niels Bohr [2] to postulate a model of atomic structure in which the atom can subsist only in certain states of well-defined energy, although transitions between such states can be induced. Transitions between two states may be associated with accompanying optical radiation, of angular frequency 0 = 2 E/h , where E is the difference in energies of the two states, and h is Planck’s constant. At this level of detail, the modern concept of atomic structure is the same as Bohr’s. Thus, accurate data on intrinsic atomic frequencies has great fundamental and practical value. For example, as noted above, such frequencies now provide the legal definition of the second. NIST has for many years maintained a program to generate, evaluate, and maintain a database of relevant atomic spectral properties; that effort is summarized elsewhere in this volume. Bohr’s basic idea, that transitions between atomic states are associated with radiation of a definite frequency, needs to be broadened somewhat to describe phenomena actually encountered in the laboratory. For example, if one prepares a sample of atoms in a highenergy state, they may make transitions to a lowerenergy state by emission of optical radiation. This process will take place over some period of time. Under simplifying but widely applicable assumptions, Viktor Weisskopf and Eugene Wigner [3] showed that quantum mechanics describes the time dependence of the intensity of emitted radiation, I (t ), as following the law of exponential decay,