Einstein coefficient of induced absorption

The probability of absorption given by the Einstein coefficient of induced absorption Bm can be expressed in terms of M 2,... 

The constant factor B 2 is the Einstein coefficient of induced absorption. Each absorbed photon of energy hv decreases the number of photons in one mode of the radiation field by one. 

The constant factor 12 is the Einstein coefficient of induced absorption. It depends on the electronic structure of the atom, i.e. on its electronic wave functions in the... 

With the definition (2.55) for the dipole matrix element Djjj, the Einstein coefficient of induced absorption Ej E finally becomes... 

Bnm = Einstein coefficient of Absorption Bmn = Einstein coefficient of Induced Emission Amn = Einstein coefficient of Spontaneous Emission (from one pole to the other in the order given)... 

Einstein obtained coefficients for induced absorption B , induced emission Bu i, and spontaneous emission Au, of light by the following thermodynamic arguments, based on Arrhenius 116 law. 

Inasmuch as a thoroughly satisfactory quantum-mechanical theory of systems containing radiation as well as matter has not yet been developed, we must base our discussion of the emission and absorption of radiation by atoms and molecules on an approximate method of treatment, drawing upon classical electromagnetic theory for aid. The most satisfactory treatment of this type is that of Dirac,1 which leads directly to the formulas for spontaneous emission as well as absorption and induced emission of radiation. Because of the complexity of this theory, however, we shall give a simpler one, in which only absorption and induced emission are treated, prefacing this by a general discussion of the Einstein coefficients of emission and absorption of radiation in order to show the relation that spontaneous emission bears to the other two phenomena. 

The UV-VIS spectrum, usually an A or log A vs. plot, in a first approximation reflects the discrete electronic states as absorption maxima at different v ,ax positions which are correlated with the molecular structure and geometry. The extinction coefficient e or more likely the integral absorption / e(v) dv, which is approximately the product of times the halfwidth AVi/2> gives information on the transition dipole moment or the Einstein coefficient of absorption or induced emission B n, which are interrelated by... 

Let us consider a molecule and two of its energy levels E) and f 2- The Einstein coefficients are defined as follows (Scheme B2.2) Bn is the induced absorption coefficient, B2i is the induced emission coefficient and A21 is the spontaneous emission coefficient. 

At low pressure, the only interactions of the ion with its surroundings are through the exchange of photons with the surrounding walls. This is described by the three processes of absorption, induced emission, and spontaneous emission (whose rates are related by the Einstein coefficient equations). In the circumstances of interest here, the radiation illuminating the ions is the blackbody spectrum at the temperature of the surrounding walls, whose intensity and spectral distribution are given by the Planck blackbody formula. At ordinary temperatures, this is almost entirely infrared radiation, and near room temperature the most intense radiation is near 1000 cm". ... 

P+ and P are the probabilities for absorption and emission, respectively B+ and B are the coefficients of absorption and of induced emission, respectively A- is the coefficient of spontaneous emission and p v) is the density of radiation at the frequency that induces the transition. Einstein showed that B+ = B, while A frequency dependence, spontaneous emission (fluorescence), which usually dominates in the visible region of the spectrum, is an extremely improbable process in the rf region and may be disregarded. Thus the net probability of absorption of rf energy, which is proportional to the strength of the NMR signal, is... 

Here, ijJ v)) is the mean and angle averaged value of the local radiation field, weighted with the profile function of the local absorption coefficient. The Aij and Bij are the Einstein coefficients for spontaneous and induced transitions, while denotes the probability for a collisional transition from state j —> i. Accordingly, the first row in eq. (10.20) accounts for spontaneous emission and collision of the molecule considered with H2, whereas in the second row induced emission processes are described. This system of rate equations has to be solved simultaneously with the generalized radiative transfer equation for every point in physical and velocity space. 

Einstein coefficients Coefficients used in the quantum theory of radiation, related to the probability of a transition occurring between the ground state and an excited state (or vice versa) in the processes of induced emission and spontaneous emission. For an atom exposed to electromagnetic radiation, the rate of absorption is given by... 

R = Bp, where p is the density of electromagnetic radiation and Bis the Einstein B coefficient associated with absorption. The rate of induced emission is also given by Bp, with the coefficient B of induced emission being equal to the coefficient of absorption. The rate of spontaneous emission Is given by A, where A is the Einstein A coefficient of spontaneous emission. The A and B coefficients are related byA = 8nhv B/( , where h is the Planck constant, v is the frequency of electromagnetic radiation, and c is the speed of light. The coefficients were put forward by... 

Since we are interested mainly in the roots of spontaneous emission, we shall quantize the electromagnetic field because we know that the semi-classical description where the atom is quantized and the field is classical, does not provide aity spontaneous emission it is introduced phenomenologically by a detailed balance of the population of the two-states atom and comparison with Planck s law. This procedure introduced by Einstein gave the well-known relationship between induced absorption (or emission), and spontaneous emission probabilities, the B12, B21 and A21 coefficients, respectively, but caimot produce the coherent aspect and its link with spontaneous emission. 

This chapter deals with basic considerations about absorption and emission of electromagnetic waves interacting with matter. Especially emphasized are those aspects that are important for the spectroscopy of gaseous media. The discussion starts with thermal radiation fields and the concept of cavity modes in order to elucidate differences and connections between spontaneous and induced emission and absorption. This leads to the definition of the Einstein coefficients and their mutual relations. The next section explains some definitions used in photometry such as radiation power, intensity, and spectral power density. 

Regarding transitions from ground to excited state and vice versa, the interaction of the radiation with the molecules are in principle the same. For this reason, the Einstein coefficients, which are a measure of probability of the two transitions, are the same for both (induced) absorption and induced emission [II], (12). (I4J. [15]. Which of the two transitions is more effective for the interaction with radiation depends only on the relative distribution of the molecules between the two states. 

Determine the Einstein coefficient Boi for the excitation of i) from the ground-state with to = oo and with the statistical weights go = 1 and gi = 3. At which spectral energy density p is the induced absorption rate equal to the spontaneous decay rate of level i) What is the intensity of a laser with a bandwidth of 10 MHz at this radiation density ... 

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