Einstein coefficient stimulated emission/absorption

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. 

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. 

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... 

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... 

The relation between the Einstein coefficients A and By is A — Snhc0v3 By. The Einstein stimulated absorption or emission coefficient B may also be related to the transition moment between the states i and j for an electric dipole transition the relation is... 

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

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 s theory involves three coefficients, and are defined for spontaneous emission ( 4 ), absorption Bji), and stimulated emission (fluorescence) Bij) (Figure 5). Ay is the probability that an atom in state will spontaneously emit a quantum hv and pass to the state j . The unit for is s . The other two coefficients are more difficult to define, since the probability of an absorption or fluorescence transition will depend on the amount of incident radiation. The transition probability in these cases is obtained by multiplication of the appropriate coefficient and the transition density at the frequency corresponding to the particular transition (Pv). and By have the unit s . The radiation density (Pv) is defined as energy per unit volume, and it has the units erg cm or gcm s The unit of Bji and By is then cmg ... 

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 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) ... 

Our discussion of Einstein coefficients and oscillator strengths in Sections 8.3 and 8.4 yields fundamental relationships among absorption coefficients, luminescence lifetimes, and probabilities for stimulated emission. The latter process is responsible for light amplification by stimulated emission (laser action), and these relationships figure prominently in the derivation of lasing criteria in Chapter 9. 

The Einstein coefficients prove to be useful for understanding the relationships among the probabilities for spontaneous emission, stimulated emission, and absorption. They are thus valuable for understanding the criteria for achieving laser action, where the competition between spontaneous and stimulated emission in the laser medium is crucial. The Einstein coefficients also lead to important insights into the relationships between the absorption and fluorescence properties of molecules, relationships that are often taken for granted in the chemical physics literature. 

Hence, the Einstein coefficients for absorption, spontaneous emission, and stimulated emission are all simply related. The factor that enters in the spontaneous emission coefficient (Eq. 8.35) has had historical importance in the development of lasers, since it implies that spontaneous emission competes more effectively with stimulated emission at higher frequencies. High-frequency lasers have therefore been more difficult to construct. This is one of the reasons why X-ray lasers have only recently been built, and why the first laser was an ammonia maser operating on a microwave umbrella-inversion vibration rather than a visible laser. 

Radiative decay is the inverse of absorption and requires coupling of the two states via either an incident photon, which results in stimulated emission, or the constantly fluctuating radiation field, which results in spontaneous emission. The radiative lifetime is also the inverse of the Einstein A coefficient. This can be calculated from the integrated molar absorption coefficient in the absorption spectrum using the Strickler-Berg relationship [55]. 

A simple relation will be derived between the probabilities for spontaneous and stimulated emission and absorption of radiation using well-known statistical distribution laws. Consider a system such as that illustrated in Fig.4.3 with two energy levels, E and E2, populated by and N2 atoms, respectively. Three radiative processes can occur between the levels, as discussed above. In the figure the processes are expressed using the so-called Einstein coefficients 6 2, B21 and A21, which are defined such that the rate of change in the population numbers is... 

FIGURE 15.27 (a) Stimulated absorption, which defines Einstein s coefficient B. (b) Spontaneous emission, which defines Einsteins coefficient A. (c) Stimulated emission, which defines Einstein s coefficient B. In stimulated emission, the two photons have the same wavelength and phase, as indicated. 

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 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... 

Equations 3.77 and 3.80 relate the stimulated emission rate and the spontaneous emission rate to the absorption coefficient, respectively. With the help of the same equations, the stimulated emission rate and the spontaneous emission rate can be related to each other. The implications of these equations are that both emission rates can be determined if the absorption coefficient along with its energy (or wavelength) dependence is known. Fortunately, the absorption coefficient is a measurable quantity. Therefore, once measured, the emission rates can be determined. The absorption rate can also be calculated with numerical techniques. Because the absorption coefficient, spontaneous emission rate, and stimulated emission rate can all be determined with the knowledge of Einstein s B coefficient (recall that the A coefficient can be calculated from the B coefficient), calculation of the B is sufficient to determine the absorption coefficient (the last part of Equation 3.78), spontaneous emission rate (Equation 3.79), and the stimulated emission rate (Equation 3.72 as A and B coefficients are related). A succinct description of the calculations leading to Einstein s B coefficient and/or the two emission rates is given below. The B coefficient represents the interaction of electron in the solid with the electromagnetic wave, which requires a quantum mechanical treatment. For further details, the reader is referred to [29]. 

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. 

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. 

---