expresses a simple idea: the instantaneous probability of decay of a given atom is independent of the time t at which it occurs. Exponential decay is encountered in a number of physical systems, such as the natural transmutation of radioactive elements.

An alternative mode of observation of exponential decay is the resolution of its frequency spectrum. This is a straightforward and convenient practice in optical spectroscopy, where the various frequency components of radiation may be spatially dispersed by reflecting it from a diffraction grating, or by passing it through a prism, and projecting it onto a screen, photographic plate, or detector array. By this means, the spectrum of a white light source is spread out into the familiar rainbow pattern. On the other hand, the spectrum of an atomic radiation governed by the decay law of Eq. (1), exhibits the "Lorentzian" profile, or line shape,

where I(w) is the intensity at the angular frequency w, and wo, as stated above, is proportional to the difference in energy between atomic states. In this representation, I plays the role of the "width" of the spectral feature. This expresses Heisenberg's uncertainty principle: the energy of a state is indeterminate by an amount inversely proportional to its lifetime. For most optical spectra generated by atoms, the ratio Two is of the order 10. Thus the human eye would perceive the spectrum of Eq. (2) as being a remarkably pure (monochromatic) and bright color, localized at the appropriate position on the viewing screen. However, if examined in detail, atomic spectra can exhibit significant deviations from the form of Eq. (2). The differences are due to the actions of physical processes neglected in the simplifying assumptions mentioned above.

Fano's 1961 treatment deals with one of the more important classes of such processes: those in which there are alternative pathways for a transition between atomic states. In quantum mechanics, the existence of alternative pathways invariably gives rise to the phenomenon of interference: the probability amplitudes associated with travel along the different paths combine with a phase relationship, just as the crests and troughs of classical wave motion can combine to yield a null displacement. In the model considered by Fano, which has been found to have broad applicability, the interference phenomenon inevitably gives rise to a "dark spot," i.e., the spectral intensity vanishes at a particular frequency. The line shape formula derived by Fano, in his chosen notation, takes the form

where

2(w-wo)/T is the angular frequency, measured from the line center wo in units of Ã, and q is a parameter that describes the interference between pathways. When q is much greater than 1, one of the pathways has the predominant transition amplitude, and Eq. (3) becomes equivalent to Eq. (2) in the region where the signal is strongest.

Fano was first motivated to understand this phenomenon 25 years previously, as a student of Enrico Fermi in Rome. Fermi suggested to Fano that, as a research project, he should try to understand photoabsorption spectra of the noble gases that had recently been observed by Hans Beutler of the University of Berlin [4].

Beutler's spectra exhibited broad, highly asymmetric absorption series, quite uncharacteristic of atomic spectra known at the time. In contrast to the wellseparated, sharp features visible in Fig. 1, Beutler's spectra showed structures whose width was comparable to their separation.

The origin of this phenomenon was correctly proposed by Beutler to be associated with the phenomenon of autoionization, which had previously been identified in atomic spectra by Ettore Majorana [5] and Allen Shenstone [6].

Autoionization occurs upon excitation of an electronic configuration that has an energy higher than that needed to ionize (i.e., remove one electron from) the atom. For example, two atomic electrons may be excited, with a total energy that is sufficient to access an alternative configuration, in which one of the electrons relaxes to a state of lower energy, while the other escapes from the atom. This phenomenon is analogous to a possible, though we hope unlikely, rearrangement of the Solar System, in which the energy released by a contraction of the orbit of Jupiter could be used to liberate the Earth from its orbit about the Sun. In the simplest case of autoionization of a state of two excited electrons, the two interfering quantum-mechanical pathways may be envisaged as follows. One involves the direct ejection of one electron from the atom, leaving behind a positively-charged ion; the other generates a state of two excited electrons, which then relaxes by energy exchange to yield the same final state, a free electron and a positive ion.

In a paper published in 1935 [7], Fano showed that spectral profiles of the type observed by Beutler could indeed be generated by quantum mechanical interference in autoionization, and he presented a formula equivalent to that of Eq. (3). However, his presentation did not involve a quantitative analysis of Beutler's data, which was quite complicated, and still, in its time, somewhat of an anomaly in atomic spectroscopy. When Fano returned to this class of problems in 1961, key developments in experimental technique had elevated it to a frontier research area of atomic physics.

The new experimental situation of the 1960s evolved along two fronts, on both of which NIST made pioneering contributions that would be adopted world-wide: high-resolution electron spectroscopy, and the application of synchrotron radiation.

John Simpson and Chris Kuyatt led the electron spectroscopy effort in the Electron Physics Section at NIST. They designed electron current sources and detectors that enabled electron scattering by atoms to be investigated in unprecedented detail [8]. Among the signal achievements of this technique was the identification of numerous "negative ion resonances" analogous to the doubly-excited electronic states encountered in photoabsorption. An electron colliding with an atom can lose energy by exciting an atomic electron, and be captured by the field of the residual ion. This can give rise to a transient state in which the incident and excited electrons orbit the positively charged ion core, which subsequently decays by the capture of one electron and the release of the other. This situation provides for interference of two quantum-mechanical amplitudes: one for direct scattering of the incident electron, the other involving the temporary capture/excitation process, which results eventually in a scattered electron. The experimental data treated explicitly by Fano in his 1961 paper were in fact provided by such an experiment. The formation of these resonances was mediated by correlation of electronic motions, a phenomenon not encompassed by the traditional "mean field" concepts of atomic structure. The study of such resonances therefore provided insight into unexplored avenues of atomic and molecular physics, which developed as a major research theme in the 1960s with interest extending to the present day.

In the early 1960s the emergence of synchrotron radiation sources greatly expanded the range of optical spectroscopy. At NBS, which played a leading role in the field, the development of a synchrotron radiation source was an offshoot of a program of investment in betatron electron accelerators which started in the late 1940s. The betatrons were originally obtained to provide high-energy electron beams for the production. of x rays. These accelerators also yielded, at first as an unappreciated byproduct, a broad band of synchrotron radiation spanning the electromagnetic spectrum from the radio to the x-ray domains. This broad-band capability was not matched by any other laboratory-based radiation source; in particular, synchrotron radiation provided exceptional coverage of the far ultra-violet spectral region, i.e., radiation with wavelengths between about 5 nm and 150 nm, vs the 500 nm wavelength characteristic of visible light. Far ultraviolet radiation has sufficient energy to cause multiple electron excitation in all elements, so that electronic states of the type observed by Beutler would become ubiquitous rather than exceptional.

Robert Madden was hired by NBS to head the Far Ultraviolet Physics Section and was charged with converting one of the betatron machines into a dedicated source of synchrotron radiation for far ultraviolet spectroscopy and radiometry, the Synchrotron Ultraviolet Radiation Facility (SURF). In collaboration with Keith Codling, David Ederer, and others, Madden made numerous pioneering explorations of this spectral region. The first major results of this work are shown in Fig. 2: the absorption spectra of helium, neon, and argon gas [9]. The helium spectra were particularly striking, because they confounded initial theoretical expectations that two series of lines would be observed, rather than the single one displayed here. A companion publication by John Cooper, Fano, and Francisco Prats [10] showed that effects of electron correlation would tend to favor strongly transitions to one of the two series. This result hinted at the existence of characteristic features of two-electron dynamics that might be found in other spectra; this became a prominent theme in atomic physics during the next two decades and is still of interest today.

Fig. 2. Absorption spectra of helium, neon, and argon atoms in the extreme ultraviolet spectral region, from [9]. These are images of photographic plates exposed to radiation from the NBS electron synchrotron (now the Synchrotron Ultraviolet Radiation FacilitySURF). The synchrotron radiation was passed through a gas cell and then dispersed by a diffraction grating to show the dependence of absorption upon wavelength (which is indicated in Ångstrom units: 1 Å = 10-10 m). Increased blackness indicates increased absorption by the gas. Note that in contrast to Fig. 1, these spectral lines clearly exhibit asymmetric profiles.

An example [11] of the application of the Fano profile formula to the synchrotron radiation data is shown in Fig. 3. The quantitative determination of the parameters q, wo, and I that are obtained from this type of analysis is essential for comparison of experiment and theory, or for comparison of experiments of two different types. For example, Fig. 3 describes an atomic state excited by photoabsorption; the same state can be excited by other means—in fact, it was this very same state, produced by electron-impact excitation, that was analyzed in Fano's 1961 paper-but the values of wo and I should be independent of the mode of excitation since they are intrinsic characteristics of the excited state.

Ugo Fano, who did his graduate work in Italy under Fermi in the early 1930s, was hired by NBS in 1946 with a charge to build up the fundamental science base of the x-ray program. In his 19 years at NBS he provided guidance and inspiration to many of the Bureau's physicists and chemists. After moving to the University of Chicago in 1965, he led his graduate students in the detailed analysis of noble-gas photoabsorption spectra. The analysis of these spectra was a noteworthy achievement of multichannel quantum defect theory, developed by Fano and coworkers along lines laid out by Michael Seaton. This theory had a pronounced influence on high-resolution laser spectroscopy in the 1970s and

1980s; its development is summarized in two articles in the February 1983 issue of Reports on Progress in Physics [12].

At NIST, the Fano profile formula evokes memories of a remarkably productive era of atomic and electron physics, one in which there was strong interplay between theory and experiment, as well as between electronic and optical spectroscopy. Many legacies of this era are visible in NIST programs today. For example, the Electron Physics Section spawned the topografiner project, work on resonance tunneling in field emission, and the development of spin-polarized electron sources and detectors—all of which are described elsewhere in this volume. The success of the SURF synchrotron source inspired to the worldwide development of synchrotron radiation as a research tool. SURF has since gone through two major upgrades and today serves as the nation's primary standard for source-based radiometry over a wide region of the optical spectrum. Fano's theory of spectral line shapes continues to be applied to a wide range of physical problems: his 1961 paper was cited over 150 times in the scientific literature in 1999.

Prepared by Charles W. Clark.

Bibliography

[1] U. Fano, Effects of Configuration Interaction on Intensities and Phase Shifts, Phys. Rev. 124, 1866-1878 (1961).

[2] N. Bohr, On the constitution of atoms and molecules, Philo. Mag. 26, 1-25 (1913).

[3] V. Weisskopf and E. Wigner, Berechnung der natürlichen Linienbreite auf Grund der Diracschen Lichttheorie, Z. Phys. 63, 54-73 (1930).

[4] H. Beutler, Über Absorptionsserien von Argon, Krypton und Xenon zu Termen zwischen den beiden Ionisierungsgrenzen 2p 3/2 und 2P1/2, Z. Phys. 93, 177-196 (1935).

[5] Ettore Majorana,Teoria dei tripletti p' incompleti, Nuovo Cimento, N. S., 8, 107-113 (1931).

[6] A. G. Shenstone, Ultra-ionization potentials in mercury vapor, Phy. Rev. 38, 873-875 (1931).

[7] Ugo Fano, Sullo spettro di assorbimento dei gas nobili presso il limite dello spettro d'arco, Nuovo Cimento, N. S., 12, 154-161 (1935).

[8] C. E. Kuyatt, J. Arol Simpson, and S. R. Mielczarek, Elastic resonances in electron scattering from He, Ne, Ar, Kr, Xe, and Hg, Phy. Rev. 138, A385-A399 (1965).

[9] R. P. Madden and K. Codling, New autoionizing atomic energy levels in He, Ne, and Ar, Phys. Rev. Lett. 10, 516-518 (1963). [10] J. W. Cooper, U. Fano, and F. Prats, Classification of twoelectron excitation levels of helium, Phys. Rev. Lett. 10, 518-521 (1963).

[11] R. P. Madden and K. Codling, Two-electron excitation states in helium, Astrophys. J. 141, 364-375 (1965).

[12] U. Fano, Correlations of two excited electrons, Rep. Prog. Phy. 46, 97-165 (1983); M. J. Seaton, Quantum defect theory, Rep. Prog. Phy. 46, 167-257 (1983).

This book [1] was written at an important point in the development of applications of electromagnetic (radio) waves to communications, navigation, and remote sensing. Such applications require accurate propagation predictions for a variety of path conditions, and this book provides the theoretical basis for such predictions. The book is based on fundamental research in electromagnetic wave propagation that James R. Wait performed in the Central Radio Propagation Laboratory (CRPL) of NBS from 1956 to 1962. The mathematical theory in the book is very general, and the "stratified media" models are applicable to the earth crust, the troposphere, and the ionosphere. The frequencies of the communication, navigation, and remote sensing applications treated in this book range all the way from extremely low frequencies (ELF) to microwaves.

The mathematical theory of electromagnetic wave propagation is based on Maxwell's equations [2], formulated by James Clerk Maxwell in the 1860s. Experimental propagation studies in free space [3] and over the earth [4] also go back over 100 years. Research in radio science, standards, and measurements began in NBS in the early 1900s, and the long history of radio in NBS has been thoroughly covered by Snyder and Bragaw [5]. CRPL was moved to Boulder in 1954, and Wait joined the organization in 1955.

The mathematics of electromagnetic wave propagation in stratified (layered) media is very complicated, and progress in propagation theory in the early 1900s was fairly slow. Wait's book [1] included the most useful theory (much of which he developed) and practical applications that were available in 1962. A hallmark

of the book is that the theory was sufficiently general that it has served to guide further theoretical and experimental propagation research to this day. Continuing demand for the book led IEEE Press to reissue it in 1996.

Chapter I provides a general introduction to the book. Chapter II is fundamental to the entire subject because it covers reflection of electromagnetic waves from horizontally stratified media. The incident field can be either a plane wave (as from a distant source) or a spherical wave (as from a nearby antenna). The theory utilizes a novel iterative approach that matches the electric and magnetic field boundary conditions at each layer interface. Then the entire layered medium can be replaced by a single interface with an equivalent surface impedance (ratio of tangential electric and magnetic fields). This approach has been found very useful by many other researchers in simplifying the analysis of complex layered geometries [6], such as printed circuit boards. Chapter III treats the case where the electromagnetic properties of the reflecting medium vary smoothly, rather than discontinuously as in a multi-layered medium. For some special profiles, such as linear or exponential, the solutions can be given exactly in terms of known mathematical functions. This type of treatment has been particularly useful in obtaining solutions for reflection of pulses from the ionosphere [7].

Whereas the methods in Chapters II and III are exact, approximate methods are developed in Chapter IV. These methods are less accurate, but they have the advantages of simplicity and physical interpretation. In general, these methods track the fields in a manner similar to ray tracing. The approximate methods have been found to be particularly useful for calculating reflections from complex ionospheric profiles [8].

Chapter V brings curved boundaries into the theory by treating wave propagation along a spherical surface. This theory is particularly important for analyzing ground-wave propagation along the surface of the earth [9]. The most important effect that can be predicted using this theory is the rapid attenuation of field strength beyond the shadow boundary. For example, the extent of daytime AM radio coverage can be determined if the ground properties are known. The most general theory can make use of the impedance boundary condition developed in Chapter II to predict field strength as a function of distance even when the curved earth model includes layering [10].

When the ionosphere is added to the curved earth model, mode theory is required to calculate field strength in the earth-ionosphere waveguide. Chapter VI presents a self-contained treatment of mode theory

that starts with very complex mathematics and ends with fairly simple and convenient approximations. The formulations are very general, and the transmitting antennas can be electric or magnetic dipoles of arbitrary orientation. At low frequencies, the earth-ionosphere waveguide acts like a cavity that can resonate [11] and enhance the background noise (typically due to lightning).

The very low frequency (VLF) band is particularly useful for long distance communication and navigation [12] because of the low attenuation rates of the earthionosphere waveguide modes. Chapter VII presents VLF approximations that are particularly useful in simplifying the general mode theory of Chapter VI. Numerous attenuation and phase velocity calculations are presented for use in communication and navigation system calculations, and the effects of earth or sea water conductivity are also included. Antenna height is also shown to be important.

The ionosphere is an ionized medium, or plasma. The earth's magnetic field alters the electromagnetic properties of the ionosphere, in particular making it anisotropic. Chapter VIII covers electromagnetic propagation effects encountered in such a magneto-plasma. Again both layering and curvature are taken into account, and practical VLF propagation effects are analyzed in detail. These results have direct application to navigation systems [13].

Chapter IX contains extensive comparisons of theory and measurement at VLF. Interesting effects such as direction of propagation (important because of the earth's magnetic field) and time of day are shown by both theory and experiment [14]. The sources can be either antennas or lightning strikes, and attenuation rates of theory and experiment are generally in good agreement [15].

Chapter X continues in the spirit of Chapter IX except that it covers extremely low frequencies (ELF). Different mathematical approximations are required to fit this lower frequency range (1 Hz to 3000 Hz) even though propagation is still primarily due to a waveguide mode. At this frequency range, the fields can penetrate sea water allowing communication with submarines [16]. Good agreement between theory and experiment [17] is again obtained.

The relationship between waveguide modes and rays is clearly developed in Chapter XI. This provides very useful insight because the ionosphere "sky wave" is often interpreted as an ionospheric reflection for a particular ray path. The mathematical results are sufficiently general that they can be applied to a variety of layered media problems [18].