The interaction of radiation and matter is discussed on the basis of Einstein's theory. The distribution of energy in an enclosure containing quantized particles is indeterminate unless there is thermal or other non-radiative interaction between the particles; the latter process is therefore essential to the production of the black-body distribution. Fluorescence cannot lower the entropy of the system.Exciting power for fluorescence.---No sharp distinction can be drawn between thermal emission and fluorescence. If ${\ensuremath{\psi}}_{\mathrm{yx}}$ is the fluorescence of frequency ${\ensuremath{\nu}}_{y}$ excited by radiation of frequency ${\ensuremath{\nu}}_{x}$ and if ${\ensuremath{\alpha}}_{x}$, ${\ensuremath{\alpha}}_{y}$, ${u}_{x}$, ${u}_{y}$ are the corresponding absorption coefficients and black-body densities, respectively, then on certain assumptions ${\ensuremath{\psi}}_{\mathrm{yx}}=\frac{k{\ensuremath{\alpha}}_{x}{\ensuremath{\alpha}}_{y}{u}_{y}}{{\ensuremath{\nu}}_{x}}$, $k$ being independent of ${\ensuremath{\nu}}_{x}$ and ${\ensuremath{\nu}}_{y}$. Much more general assumptions lead to: $\frac{{\ensuremath{\psi}}_{\mathrm{yx}}}{{\ensuremath{\psi}}_{\mathrm{xy}}}=\frac{{\ensuremath{\nu}}_{y}{u}_{y}}{{\ensuremath{\nu}}_{x}{u}_{x}}$. These equations ought to apply to gas fluorescence, and it is quite possible that the latter may hold generally for liquids and solids and the former in those cases where the shape of the fluorescence band does not vary with the exciting frequency. The equations agree with the few data in existence.
The interaction of radiation and matter is discussed on the basis of Einstein's theory. The distribution of energy in an enclosure containing quantized particles is indeterminate unless there is thermal or other non-radiative interaction between the particles; the latter process is therefore essential to the production of the black-body distribution. Fluorescence cannot lower the entropy of the system.Exciting power for fluorescence.---No sharp distinction can be drawn between thermal emission and fluorescence. If \ensuremathψyx is the fluorescence of frequency \ensuremathνy excited by radiation of frequency \ensuremathνx and if \ensuremathαx, \ensuremathαy, ux, uy are the corresponding absorption coefficients and black-body densities, respectively, then on certain assumptions \ensuremathψyx=\ensuremathνxk\ensuremathαx\ensuremathαyuy, k being independent of \ensuremathνx and \ensuremathνy. Much more general assumptions lead to: \ensuremathψxy\ensuremathψyx=\ensuremathνxux\ensuremathνyuy. These equations ought to apply to gas fluorescence, and it is quite possible that the latter may hold generally for liquids and solids and the former in those cases where the shape of the fluorescence band does not vary with the exciting frequency. The equations agree with the few data in existence.
Suppose that the gas is in equilibrium with a radiation field B(?, T). In a steady state, the number atoms leaving the lower state must equal the number leaving the upper state. Combining the last two equations, The probability that an atom will be in a state with energy Eu is exp{- Eu/kT) . This is proved in any book on statistical mechanics, but it is beyond the scope of this chapter. The relative populations of the two states is Using equation (33), Therefore, combining equations (37) and (38), Since the constants and depend only on the structure of the atom, and not on temperature or other aspects of the environment,25 Then So, using the Plank black-body law [equation (2)], These coefficients are called the Einstein coefficients; they depend only on the nature of the atom and not on the temperature, the ambient radiant intensity, or any other aspect of the atom’s environment. Therefore, equations (42) and (44) are true in general. In particular, note that this argument shows that stimulated emission must exist; i.e., . This proved the existence of stimulated emission long before it had been observed in the laboratory; today, of course, it is the basis of the laser (Light Amplification by Stimulated Emission of Radiation) industry. Stimulated emission is necessary, as I have already mentioned, to keep the Second Law of thermodynamics from being violated, because without it, atomic absorption and emission processes could not be in balance with emission from a black body. If there is an enclosure with perfectly black walls with temperature T, then the radiation field inside the enclosure must be equal to the Planck function B(?, T). But now introduce a gas into that enclosure. Without stimulated emission, there would be a continuous transfer of energy either to or from the gas. This, as we know, is impossible.
Most introductory courses on physics mention the basics of atomic physics: we learn about the ground state and excited states, and that the excited states have a finite life-time decaying eventually to the ground state while emitting radiation. You might have hoped for a better understanding of these decay processes when you started studying quantum mechanics. If this was the case, you must have been disappointed. Courses on basic quantum mechanics in fact assume that all quantum states have an infinitely long life-time. You learn how to employ the Schrödinger equation to determine the wave function of a particle in various setups, for instance in an infinitely deep well, in a parabolic potential, or in the spherically symmetric potential set up by the nucleus of a hydrogen atom. These solutions to the Schrödinger equation are explained as being stationary, the only time-dependence of the wave function being a periodic phase factor due to the energy of the state. Transitions between different quantum states are traditionally not discussed and left for courses on advanced quantum mechanics. If you have been disappointed by this, then this chapter will hopefully provide some new insight: we learn how matter and radiation interact, and, in particular, how excited atomic states decay to the ground state. Our main tool to calculate emission and absorption rates for radiative processes is Fermi’s golden rule (see Section 1.6.1). We thus start by refreshing the golden rule and specifying its proper use. Before we turn to explicit calculations of transition rates, we use the first part of the chapter for some general considerations concerning emission and absorption, those being independent of the details of the interaction between matter and electromagnetic radiation. In this context we briefly look at master equations that govern the probabilities to be in certain states and we show how to use those to understand properties of the equilibrium state of interacting matter and radiation. We show that it is even possible to understand so-called black-body radiation without specifying the exact interaction Hamiltonian of matter and radiation.
Introduction The preceding chapters have focused on atomic transitions induced by applied radiation. That is, there is an optical field illuminating the atoms, and their response is calculated. It is implicitly assumed that an excited atom left alone will spontaneously return to the ground state, and conserve energy by emitting light that satisfies E 2 − E 1 = ħ ω 21 in accordance with the Planck and Bohr pictures. But this spontaneous emission process is not addressed quantitatively in these earlier chapters. Spontaneous emission is one of the most pervasive phenomena of atomic physics, and yet its origins remain among the least well-understood. The number of misconceptions is enormous, and they are to be found in textbooks, journal articles, and lecture notes. Because the natural decay of an excited atom adds energy to what is usually an empty mode of the electromagnetic field, the process is fundamentally quantum mechanical. Classical or even semi-classical descriptions cannot be assured to give correct answers. This is not to say that such discussion is to be avoided, but only that caution is needed in interpreting the results. Toward the end of the nineteenth century, many experimental results made it clear that classical mechanics needed to be supplemented by new theories. The first venture into the new physics was Planck's hypothesis that the energy of classical radiating oscillators like those discussed in Chap. 1 would have to be quantized into integer multiples of ħ ω. This notion led to his famous formula for the spectrum of thermal radiation from a non-reflective object (black body), and this formula describes a radiation field in thermodynamic equilibrium with its environment. The result of Planck's hypothesis in the context of thermal equilibrium plays a vital role in the discussion of spontaneous emission. Einstein A - and B -coefficients Soon after Bohr proposed a model of atomic structure consisting of discrete energy levels, connected by radiation whose energy satisfied Δ E = ħω when atoms made transition between them, Einstein published the first serious attempt to connect the absorption and emission processes. Absorption, he reasoned, could occur only in the presence of applied radiation, whereas spontaneous decay occurred without the application of any field.
In 1916, A. Einstein developed a model of the absorption and emission of radiation from which he derived M. Planck’s radiation law. He postulated light-induced (stimulated) emission, because within the framework of classical physics an oscillator can, depending on the phase difference, both extract energy from the electromagnetic field and transfer energy to it. In a non-coherent field, α = β should hold, where α denotes the kinetic constant of absorption and β that of stimulated emission. This assumption is indispensable for deriving the radiation law from his model. In this context, it is problematic that a certain degree of field coherence follows necessarily from his model, which implies that α ≠ β. Moreover, the model cannot readily be generalised to fields of arbitrary coherence. An alternative model for the absorption and emission of radiation by matter is therefore developed here. On the one hand, it is based on the assumption that, within a blackbody, quanta of different energies may recombine freely; on the other hand, it employs the “pairwise interference model of quantum theory” (PIMOQ). The alternative model of the interaction between light and matter allows the radiation law to be derived for thermal equilibrium in a non-coherent field, under which conditions, according to the model, no stimulated emissions occur. It also describes the behaviour of systems with a coherent field (interference experiment, laser). Furthermore, considerations are presented regarding the energy distribution within a blackbody and the explanatory scope of A. Einstein’s model in this respect.
Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.
At the beginning of the century doubts were growing about the whole edifice of classical physics, including the laws of mechanics in their Lagrangian and Hamiltonian forms, of statistical thermodynamics, as formulated by Boltzmann and Gibbs, and of optics, namely Maxwell’s and Fresnel’s theory. Einstein proposed that electromagnetic radiation exists in discrete quanta whilst propagating in space as well as during the emission or absorption processes. Scattering by free electrons is a common interaction connected with the absorption of electromagnetic radiation. If the electromagnetic energy is large compared with the ionization energy, the atomic electron can be treated as free. The characteristic features of the scattered radiation have been explained by Compton, on the basis of an elastic collision of a photon with a free electron, where both energy and momentum are conserved. The concept of the matter wavelength holds for both relativistic and non-relativistic particles, provided that the appropriate form of momentum is used.