Einstein spontaneous emission

According to Judd-Ofelt theory, one can evaluate the radiative lifetime of any excited state of interest via Einstein spontaneous emission coefficients. The rate of relaxation, A, from an initial state fJ) to final state if J ) through radiative processes is given by (Condon and... 

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... 

Quantum mechanics also yields an expression for the Einstein spontaneous emission coefficient (the reciprocal of the mean radiative lifetime) [8] ... 

We now make two coimections with topics discussed earlier. First, at the begiiming of this section we defined 1/Jj as the rate constant for population decay and 1/J2 as the rate constant for coherence decay. Equation (A1.6.63) shows that for spontaneous emission MT = y, while 1/J2 = y/2 comparing with equation (A1.6.60) we see that for spontaneous emission, 1/J2 = 0- Second, note that y is the rate constant for population transfer due to spontaneous emission it is identical to the Einstein A coefficient which we defined in equation (Al.6.3). 

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... 

The acronym LASER (Light Amplification via tire Stimulated Emission of Radiation) defines the process of amplification. For all intents and purjDoses tliis metliod was elegantly outlined by Einstein in 1917 [H] wherein he derived a treatment of the dynamic equilibrium of a material in a electromagnetic field absorbing and emitting photons. Key here is tire insight tliat, in addition to absorjDtion and spontaneous emission processes, in an excited system one can stimulate tire emission of a photon by interaction witli tire electromagnetic field. It is tliis stimulated emission process which lays tire conceptual foundation of tire laser. 

An important process has not been included in the analysis. It is the possibility of spontaneous emission. Were it not for such a process, in the absence of electromagnetic radiation a molecule in the excited state ro would be forced to remain there forever. Thus, in Einstein s analysis of this problem three competing processes were considered to be in equilibrium, leading to tbf expression... 

The word LASER is an acronym for Light Amplification by Stimulated Emission of Radiation. The physical process upon which lasers depend, stimulated emission, was first elucidated by Einstein in 1917 (1). Einstein showed that in quantized systems three processes involving photons must exist absorption, spontaneous emission, and stimulated emission. These may be represented as follows ... 

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. 

It is interesting to note that for a resonant transition (i.e. coinciding absorption and emission frequencies), the reciprocal of the radiative lifetime is equal to the Einstein coefficient Ai for spontaneous emission (see Box 3.2). 

A being the radiative rate (labeled in such a way because it coincides with the Einstein coefficient of spontaneous emission) and Anr being the nonradiative rate, that is, the rate for nonradiative processes. The solution of the differential equation (1.16) gives the density of excited centers at any time r ... 

Appendix A3 The Calculation of the Probability of Spontaneous Emission by Means of Einstein s Thermodynamic Treatment... 

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. 

Finally, using the relationship between the Einstein A and B coefficients (A3.8) together with the previous expression, we obtain the following expression for the probability of spontaneous emission ... 

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... 

On the basis of these formulae one can convert measurements of area, which equals the integral in the latter formula, under spectral lines into values of coefficients in a selected radial function for electric dipolar moment for a polar diatomic molecular species. Just such an exercise resulted in the formula for that radial function [129] of HCl in formula 82, combining in this case other data for expectation values (0,7 p(v) 0,7) from measurements of the Stark effect as mentioned above. For applications involving these vibration-rotational matrix elements in emission spectra, the Einstein coefficients for spontaneous emission conform to this relation. 

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". ... 

Einstein s laws of absorption and emission describe the operation of lasers. The luminescence of minerals, considered in this book, is a spontaneous emission where the luminescence is independent of incident radiation. In a stimulated emission the relaxation is accomplished by interaction with a photon of the same energy as the relaxation energy. Thus the quantum state of the excited species and the incident photon are intimately coupled. As a result the incident and the emitted photons will have the same phase and propagation direction. The emitted light of stimulated emission is therefore coherent as opposed to the... 

Albert Einstein realised that relaxation from an excited state can occur due to spontaneous emission or stimulated emission. The spontaneous emission from the exeited state occur without any interference from the enviromnent. The stimulated emission is induced by collisions with photons with energy very similar to the energy difference between the excited state and the ground state. The ratio between the probability for spontaneous emission and a stimulated emission is given by... 

From Figure 7.10 it is seen that spontaneous emission according to the Planck theory of Black body radiation as well as Einstein s work starts to dominate above 10 Hz at 300K, this corresponds to the infrared range of the electromagnetic spectram. Note, that if the temperature increases the zero crossing point moves into the visual and UV range. 

Substituting the Einstein coefficient A for spontaneous emission according to... 

Excited state lifetimes are related to the Einstein coefficients of spontaneous emission Amn and can be approximately calculated from the expression... 

II. Einstein s Theory of Spontaneous Emission in Unstable Particles. . 12... 

An early (perhaps the first) example of a quantum treatment of a dissipative process is Einstein s theory of spontaneous emission.6 To describe the interaction between matter and light, Einstein assumed the Boltzmann type of kinetic equations... 

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