Einstein Coefficient of Stimulated Emission

Stimulated emission That part of the emission which is induced by a resonant perturbing electromagnetic field. The transition between states, n and m, is governed by the Einstein coefficient of stimulated emission, Bnm- CIDNP emission and lasing action are examples of processes which require stimulated emission. 

The radiation field induces also a transition from the upper to the lower state (stimulated emi.ssion). The rate is w = B p, where B is the Einstein coefficient of stimulated emission. The rate of spontaneous emission is w" = A, with A the coefficient of spontaneous emission (note the absence of p in this expression). This is... 

The quotient of the second-order kinetic rate constant over the energy of transition is called the Einstein coefficient of stimulated emission, denoted B., and expressed in m r s Precise, quantum-mechanical calculations give the following equation for the absorption coefficient (wavenumber basis) ... 

The molecule-intrinsic factor in the intensities of emission spectra can be obtained from the well-known Einstein coefficients (see, for example. Refs. [20, 21]). For the two states i and f considered above, whose energies are Ei and Ef, respectively, with Ei < Ef, we define as the Einstein coefficient for absorption, Bfl as the Einstein coefficient for stimulated emission, and Afl as the Einstein coefficient for spontaneous emission. We denote by Ni and Nf the number of molecules with energies Ei and Ef, respectively, and the Einstein coefficients are defined such that, for example, the change in Nf caused by electric dipole transitions to and from i is given by... 

Bnm Einstein s coefficient of (stimulated) absorption Bmn Einstein s coefficient of stimulated emission Amn Einstein s coefficient of spontaneous emission... 

The constant A is the Einstein coefficient of spontaneous emission. It can be shown that the coefficients of stimulated absorption and emission are equal and that the coefficient of spontaneous emission is related to them by... 

The light emitted in the spontaneous recombination process can leave tire semiconductor, be absorbed or cause additional transitions by stimulating electrons in tire CB to make a transition to tire VB. In tliis stimulated recombination process anotlier photon is emitted. The rate of stimulated emission is governed by a detailed balance between absorjDtion, and spontaneous and stimulated emission rates. Stimulated emission occurs when tire probability of a photon causing a transition of an electron from tire CB to VB witli tire emission of anotlier photon is greater tlian that for tire upward transition of an electron from tire VB to tire CB upon absorjDtion of tire photon. These rates are commonly described in tenns of Einstein s H and 5 coefficients [8, 43]. For semiconductors, tliere is a simple condition describing tire carrier density necessary for stimulated emission, or lasing. This carrier density is known as... 

These equations are similar to those of first- and second-order chemical reactions, I being a photon concentration. This applies only to isotropic radiation. The coefficients A and B are known as the Einstein coefficients for spontaneous emission and for absorption and stimulated emission, respectively. These coefficients play the roles of rate constants in the similar equations of chemical kinetics and they give the transition probabilities. 

N, Ng...population of excited, ground states in two-level systems Bu, B. ..Einstein coefficient for absorption, stimulated emission A.Einstein coefficient for spontaneous emission... 

V is the resonator volume, c is the speed of light, rp is the photon lifetime in the cavity, a% j is the cross-section for stimulated emission, Av>v is the Einstein coefficient for spontaneous emission, and Ais the population inversion. It is assumed that emission occurs in only one v/-line in a given vibrational band, thus draining the entire population inversion of this band. 

It should be realized that absorption and emission as discussed in Chapters 2 tuid 3 are different processes, since the former needs a radiation field and the latter not. Einstein considered the problem of transition rates in the presence of a radiation field [31]. For the transition rate from lower to upper level he wrote w = Bp, where B is the Bin.stein coefficient of (.stimulated) absorption and p is the radiation density. 

In similar fashion, two other rates can be defined for the cases of stimulated emission and for absorption. These can be expressed in terms of two rate constants (or Einstein coefficients), one for stimulated emission and one for absorption. The rates are proportional to the numbers of atoms in the relevant state and the number of photons present. Thus the rate at which atoms in state Eq are excited to state El is then given by ... 

Einstein transition probability - A constant in the Einstein relation Ay +ByP forthe probability of atransition between two energy levels and y in a radiation field of energy density p. The Ay coefficient describes the probability of spontaneous emission, while By and Bj, govern the probability of stimulated emission and absorption, respectively (By =... 

The coefficients Asi, B i, and Bu. came to be known as the Einstein coefficients for spontaneous emission and for stimulated emission and absorption of radiation, respectively. The rates of spontaneous and stimulated emission and Wei are connected by the expression... 

Einstein coefficients for absorption and stimulated emission, denoted by and respectively. The expressions for B j, and Bj are then confirmed by means of quantum mechanics using time-dependent perturbation theory. This enables the probability of stimulated emission and absorption of radiation to be given in terms of the oscillator strengths of spectral lines. Finally we show that there is close agreement between the classical and quantum-mechanical expressions for the total absorption cross-section and explain how the atomic frequency response may be introduced into the quantum-mechanical results. 

Einstein derived the relationship between spontaneous emission rate and the absorption intensity or stimulated emission rate in 1917 using a thennodynamic argument [13]. Both absorption intensity and emission rate depend on the transition moment integral of equation (B 1.1.1). so that gives us a way to relate them. The symbol A is often used for the rate constant for emission it is sometimes called the Einstein A coefficient. For emission in the gas phase from a state to a lower state j we can write... 

In the above rather simplified analysis of the interaction of light and matter, it was assumed that the process involved was the absorption of light due to a transition m - n. However, the same result is obtained for the case of light emission stimulated by the electromagnetic radiation, which is the result of a transition m -> n. Then the Einstein coefficients for absorption and stimulated emission are identical, viz. fiOT< n = m rt. 

Stimulated emission. The upper state can also decay by stimulated emission controlled by the Einstein B coefficient and the intensity of photons present of the same frequency. 

In a celebrated paper, Einstein (1917) analyzed the nature of atomic transitions in a radiation field and pointed out that, in order to satisfy the conditions of thermal equilibrium, one has to have not only a spontaneous transition probability per unit time A2i from an excited state 2 to a lower state 1 and an absorption probability BUJV from 1 to 2 , but also a stimulated emission probability B2iJv from state 2 to 1 . The latter can be more usefully thought of as negative absorption, which becomes dominant in masers and lasers.1 Relations between the coefficients are found by considering detailed balancing in thermal equilibrium... 

Equation (A3.7) shows the equality between the probabilities of absorption and stimulated emission that we have already established for monochromatic radiation in Equation (5.15). Equation (A3.8) gives the ratio of tlie spontaneous to the induced transition probability. It allows us to calculate the probability A of spontaneous emission once the Einstein B coefficient is known. 

As shown in the previous chapter, the spontaneous decay rate of the n state to the lower lying n t state is given by the Einstein A coefficient nf. nf-7 In the presence of thermal radiation the stimulated emission rate Kn,fnt, is simply n times as large as the spontaneous rate. Explicitly,... 

The large Einstein radiative coefficients [225] and the widely spaced vibration-rotation quantum states make HF peculiarly prone to stimulated emission, and a large proportion of the chemical lasers which have been reported operate on lines in the infrared bands of this molecule [224], H-atom abstraction reactions by F and F-atom abstraction by H are both normally exothermic, and HF is quite generally produced in a vibrational distribution giving rise to oscillation. However, the systems are complex frequently both types of reaction occur, and the details of the vibrational distribution resulting from chemical reaction are difficult to evaluate. 

Transition probabilities, such as the Einstein spontaneous emission coefficient, Aij, axe defined so that, in the absence of collisions, nonradiative decay processes (see Chapters 7 and 8), and stimulated emission the upper level, i, decays at a rate... 

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