1Einstein coefficients, cross sections, f values, dipole moments, and all that Robert C. Hilborn Department of Physics, Amherst College, Anherst, MA 01002 The relationships among various parameters describing the strength of optical transitions in atoms and molecules are reviewed. The application of these parameters to the description of the interaction between nearly monochromatic, directional light beams and atoms and molecules is given careful attention. Common pitfalls in relating these parameters are pointed out. This is a revised (February, 2002) version of a paper that originally appeared in Am. J. Phys. 50, 982–986 (1982). I. INTRODUCTION Several parameters are commonly used to describe the strength of atomic and molecular optical transitions. The Einstein A and B coefficients, f values (also called “oscillator strengths”), and transition dipole moments are all atomic and molecular parameters related to the “strength” of the transition. In many practical situations, on the other hand, it is useful to define an absorption coefficient (or for lasers, a gain coefficient) to describe the absorption (or amplification) of a beam of light passing through a medium consisting of the atoms or molecules of interest. From a “kinetics” point of view, the absorption or scattering of radiation is described as a reaction or scattering process and the probability of absorption or scattering is given in terms of a cross section. It is the purpose of this paper to review the relationships among these descriptions of the light-matter interaction and to point out common pitfalls in using these relationships. An examination of books1-7 dealing with light-matter interaction shows a wide variety of expressions relating these parameters. Differences among these expressions involving factors of 2π, εo, etc., of course, can be traced to differing units used in the definitions of the parameters. However, these differences are often exacerbated because many different (and often not clearly defined) measures of light “intensity” are used in the definitions of the parameters. Further difficulties arise when these parameters are applied to some practical problems. The careful reader notes that the relationships among these parameters are almost always derived under the assumption that the atom is interacting with an isotropic, unpolarized, broad-band (wide frequency range) light field. This careful reader is then2wary of applying these parameters to describe the interaction between atoms and directional, polarized, nearly monochromatic light beams. Although this paper contains no new results, I believe that a unified discussion of the relationships among these parameters will prove to be of value to students, teachers, and researchers. For simplicity, the discussion will be limited to isolated atoms (e.g., atoms in a low-density gas or in an atomic beam) and to an isolated electric dipole transition between an upper level labeled 2, and a lower level labeled 1. I will use the word “atom” generically to mean the material system of interest. The index of refraction of the surrounding medium is assumed to be unity. SI units are used throughout the paper. II. ABSORPTION COEFFICIENT AND ABSORPTION CROSS SECTION Let us begin the discussion by defining a phenomenological absorption coefficient ()αω. For a beam of light propagating in the x direction, the absorption coefficient is defined by the expression 1()()()diidxωαωω=− , (1) where i(ω) is some measure of the power in the light beam at the frequency ω. (The power is actually a time-averaged quantity, averaged over several optical cycles.) In most practical applications, the frequency dependence of the absorption and emission processes is important. In this paper, it is assumed that all of these frequency dependencies can be expressed in terms of a line shape function g(ω). The normalization of g(ω) is chosen so that ()1gdωω+∞−∞=∫ . (2) (Having the lower limit of the normalization integral be −∞ greatly simplifies the normalization calculation. Negative frequencies have no special physical significance.) If the atoms are undergoing collisions (leading to collision broadening of the spectral line) or moving about (leading, in general, to Doppler broadening), then g(ω) describes the appropriate ensemble-average line shape. Note that g(ω) has the dimensions of 1/angular frequency. One of the common pitfalls lies in not explicitly recognizing which independent variable is being used to describe the frequency or wavelength dependence. In this paper, angular frequency will be used except where noted. We expect α(ω) to be proportional to the number n1 of atoms in level 1 (per unit volume) that the beam intercepts. (We assume that stimulated emission effects are negligible.8) The absorption cross section σa(ω) is then defined by 1()()anαωσω= . (3)3 As long as the light is not too intense and the atomic density not too large, both α(ω) and σa(ω) will be proportional to g(ω): 0()()gαωαω= , (4) 0()()agσωσω= . (5) Note that α0 and σ0 are the frequency-integrated absorption coefficient and cross section, respectively: 0()dααωω+∞−∞=∫ , (6) 0()dσσωω+∞−∞=∫ , (7) with dimensions of angular frequency/distance and angular frequency × area. The frequency-integrated parameters are of limited practical usefulness. They can be used to describe the absorption and scattering of a beam of light whose frequency bandwidth is large compared to the width of g(ω). In most cases, however, we are interested in the behavior of light beams whose bandwidth is on the order of, or less than, the width of g(ω). In those cases, we must use α(ω) and σa(ω). III. EINSTEIN COEFFICIENTS In 1917 Einstein introduced A and B coefficients to describe spontaneous emission and induced absorption and emission. The Einstein A coefficient is defined in terms of the total rate of spontaneous emission 21sW from an upper level 2 to a lower level 1 for a system of N2 atoms in the upper level: 21212sWAN= . (8) If level 2 can decay only by radiative emission to level 1, then A21 must be the reciprocal of the spontaneous radiative lifetime tspon of level 2: 21spon1/At= . (9) (If level 2 can decay to several lower levels, the more general relation spon21/iitA=∑ must be used, where the sum is over all energy levels to