Einstein coefficient for absorption

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. 

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... 

Consider a system in which matter and radiation are in equilibrium in a closed cavity at temperature T. (This equilibrium situation does not generally hold in spectroscopy, but the transition probabilities are fundamental properties of the interaction between radiation and matter and cannot be affected by the presence or absence of equilibrium.) As before, let be greater than (0). The probability of absorption from state n to state m is proportional to the number of photons with frequency near vmn the number of photons is proportional to the radiation density u(vmn). Hence the rate of absorption is given by Bn t,mNnu(i mn)t where Nn is the number of molecules in state n and Bn m is a proportionality constant called the Einstein coefficient for absorption. From the discussion following (3.46) and from (3.47), it follows that... 

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... 

Here +, B are the Einstein coefficients for absorptive transitions to the two exciton states, and +, are the incident light intensities at the respective transition wavelengths. (The Einstein coefficients are proportional to the extinction coefficients of the absorption bands). Throughout this chapter, the notation x indicates a unit vector that points in the same direction as x. 

The Einstein coefficient for absorption Bik is proportional to the absorption coefficient a, integrated over the absorption profile. 

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. 

B is called the Einstein coefficient for absorption. Because each transition from to Wb removes an amount of energy hvba from the radiation, the sample must absorb energy at the rate Bp(vba)Nahvba- Equation (5.3) often is presented without the factors f and n so that it refers to a sample in a vacuum. 

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. 

Comparing equation (9.41) with the definition of the Einstein coefficient for absorption given in section 9.2 leads to the result... 

Einstein coefficient of absorption for the pump wavelength calculated as B 531 In(lO) / where c is the velocity of light and is Avogadro s number. Other details of the modeling may be found in reference 3. 

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. 

The Einstein coefficient of absorption of radiation for the longitudinal (along the z axis) and the transverse (perpendicular to the z axis) electromagnetic fields can be obtained from these matrix elements and are given by... 

We now finally define the Einstein coefficient for induced absorption as ... 

The expressions for the y and z directions are similar. Thus we obtain for the Einstein coefficient of absorption B , m the equation... 

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. 

The Einstein coefficients for induced emission and absorption are identical and can be expressed as... 

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